Ramanujan笔记本中级数有限三和形式的一个新结果

摘要

我们考虑一个函数$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$具有$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="ℂ">ℂ</trans>$和$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="\mathbb{N}">\mathbb{N}</trans>$在对称域上，它等于拉马努扬第二本笔记本第三章第16条所述的无穷级数之和。由于该函数的建立与标记树理论密切相关，因此引起了新的关注。然而，据我们所知，封闭式解决方案允许快速计算$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$直到现在，在Mathematica中还没有显式使用递归。我们提出的公式根据变量变换零件$\mathrm{<trans\; data-src="u">u个</trans>}$然后出现在由二项式和第二类斯特林数组成的有限三重和中。

1.简介

斯里尼瓦萨·拉马努扬（1887-1920）的成文思想在被写下来100多年后一直是一种灵感。在[1]，提出并讨论了他在《印度数学学会杂志》上发表的问题。式（9）[1]引起了我们的注意，随后的讨论使我们[2]，Berndt、Evans和Wilson（1983）编写的Ramanujan第二本笔记本第3章的摘要/讨论。在那里，第16条规定，在收敛域上（c.f.，方程（11），第4节):

$$\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{<trans\; data-src="\infty ">\infty </trans>}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}\frac{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="k">k个</trans>}}}{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="!">!</trans>}<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}\frac{{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(1)

哪里$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="ℂ">ℂ</trans>$和$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="\mathbb{N}">\mathbb{N}</trans>$其中多项式${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$由递归定义

$${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(2)

具有${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="if">如果</trans>}<trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\notin ">\notin </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$和${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$方程（2）归于Berndt、Evans和Wilson，而在Ramanujan的原著中提供了三项等效递归。我们在文献中没有遇到任何允许快速计算$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$无需设置递归。本工作的目标是（i）提出这样一个公式（方程式（5）给出的关键结果），（ii）通过计算手段对其进行支持，以及（iii）鼓励更广泛的社区严格证明推测的身份。

在的介绍性文本中[2]，声明如下：“条目16和17在文献中似乎没有被扩展，似乎是进一步富有成果的研究的基础”。该声明也引用于[三]这是曾轶可1999年的作品，他进一步指出，自[2]，“一些作者对条目17中的序列进行了算术和组合研究”，而“对于条目16似乎什么也没有做”。曾先生注意到肖尔对标记树的研究[4]涉及多项式的递推公式${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$类似于${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$因此，它在[三]那个

$${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(3)

意味着从例如。，${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="15">15</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="35">35</trans>}$，如下${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="15">15</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="25">25</trans>}$此外，了解${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$允许我们通过方程系统通过方程（3）计算出${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$。我们应该注意到[三,5]

$${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}^{{\mathrm{<trans\; data-src="deg">度</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>$$

(4)

哪里${\mathcal{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}$有标签的树在上面吗$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\}">\}</trans>$具有$\mathrm{<trans\; data-src="t">t吨</trans>}$不正确的边缘，以及${\mathrm{<trans\; data-src="deg">度</trans>}}_{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$是树中最小节点的子节点数$\mathrm{<trans\; data-src="T">T型</trans>}$。我们在此补充了[5]带有案例的插图${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$。所有树都被认为具有四个标记节点和一个不正确的边。边缘$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans>$如果子树的最小节点根在$\mathrm{<trans\; data-src="b">b条</trans>}$（包括$\mathrm{<trans\; data-src="b">b条</trans>}$本身）的价值超过$\mathrm{<trans\; data-src="a">一</trans>}$; 否则会被归类为不当行为。我们找到了七种方法来实现具有四个标记节点和一个不正确边的树，如图1节点1在三种情况下有两个子节点，在四种情况下只有一个子节点。因此，方程式（4）表明${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="4">4</trans>}$，同意可能阅读的内容，例如[5].

所引用的作品对[2]. 在我们看来，大多数后续作品以这样或那样的方式处理[2]在标记树的理论框架中这样做。如上所述，一个显式公式允许快速计算$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$而不需要设置递归。这项工作的关键结果是推测出的恒等式（对于$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$):

$${{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}\frac{{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}}<trans\; data-src="=">=</trans>{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="r">第页</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="j">j个</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}_{\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}}{<trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}\\ \mathrm{<trans\; data-src="k">k个</trans>}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="k">k个</trans>}\\ \mathrm{<trans\; data-src="s">秒</trans>}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="\{">\{</trans>\begin{array}{c}\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}\\ \mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}\end{array}<trans\; data-src="\}">\}</trans>\frac{{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}}}$$

(5)

其中右侧的花括号用于表示第二类斯特林数。等式（5）的右侧提供了一个有效的计算$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$在Mathematica中[6]通过内置函数二项式[]和StirlingS2[]。我们在中描述第2节我们是如何在第3节，我们描述了为加强推测所做的努力。结论见第4节需要强调的是，在没有证明的情况下，方程（5）仍然是推测的。

2.方法

例如，通过针对方程式（1）左侧的简单特殊情况$\mathrm{<trans\; data-src="x">x个</trans>}$设置为小整数和$\mathrm{<trans\; data-src="u">u个</trans>}$设置为简单分数（例如$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}$,$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans><trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="etc">等</trans>}<trans\; data-src=".">.</trans>$)，我们能够提出一些函数，这些函数似乎为无限和提供了闭式解，例如。，$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}$和$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}$.熟悉之后[2]以及方程（1）和（2），我们注意到在比较我们的解的形式时存在差异。鉴于${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$-多项式呈现形式的显式解

$$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="r">第页</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="j">j个</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}\frac{{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="k">k个</trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}}}<trans\; data-src=" "> </trans><trans\; data-src=",">,</trans>$$

(6)

具有${\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}$取整数值（不完全是非负的），我们少数实验发现的解的形式如下

$$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="r">第页</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="j">j个</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}\frac{{\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}}}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}}}<trans\; data-src=" "> </trans><trans\; data-src=",">,</trans>$$

（7）

其中整数系数${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$如果$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}$否则绝对是肯定的。例如，在$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}$，我们有（通过方程式（4）检查或参见，例如[5]):${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="6">6</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="11">11</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="6">6</trans>}$;${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="6">6</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="26">26</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="26">26</trans>}$;${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="15">15</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="35">35</trans>}$;${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="15">15</trans>}$。这为

$$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="4">4</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="6">6</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="4">4</trans>}}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="6">6</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="5">5</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="11">11</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="4">4</trans>}}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="26">26</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="5">5</trans>}}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="6">6</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="6">6</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="4">4</trans>}}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="26">26</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="5">5</trans>}}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="35">35</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="6">6</trans>}}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="15">15</trans>}}{{\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}^{\mathrm{<trans\; data-src="7">7</trans>}}}<trans\; data-src=")">)</trans><trans\; data-src=" "> </trans><trans\; data-src=",">,</trans>$$

(8)

哪里${\mathrm{<trans\; data-src="u">u个</trans>}}_{<trans\; data-src="*">*</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}$.在设置${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$对于$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$1和2是一个简单的代数运算${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$,${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="6">6</trans>}$,${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="4">4</trans>}$,${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="15">15</trans>}$,${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$,${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$和${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="15">15</trans>}$。根据此示例，我们构造了${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}$的系数$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans><trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="4">4</trans>}$和$\mathrm{<trans\; data-src="5">5</trans>}$，如所示图2（顶部阵列）。

这些数组被考虑了一段时间，直到注意到从底部开始的第二行是熟悉的二项式系数。然后发现二项式“$\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}$结束$\mathrm{<trans\; data-src="k">k个</trans>}$“不仅捕获了第行的序列$\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}$，但也为数组的每个元素提供了适当的除数。结果商显示为中的底部数组图2突破性的一步是查阅在线整数序列百科全书（OEIS）[7]带有查找连续出现的3、10、25和56的查询（在中标记为红色图2). 这使我们找到了序列A000247，其中还对序列A000478进行了交叉引用，从15、105、490开始（作为图2)和A058844，以105、1260开头（作为图2). 这表明，通过对OEIS的描述，我们构建的数组的条目对应于放置方式的数量$\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="k">k个</trans>}$标签项目放入$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}$不可区分的盒子，使得每个盒子中至少放置2个物品。出租$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans><trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans>$表示放置方式的数量$\mathrm{<trans\; data-src="a">一</trans>}$标签项目放入$\mathrm{<trans\; data-src="b">b条</trans>}$我们进一步从中了解到，每个盒子中至少放置了两件物品[7]（通过对序列A008299的注释）

$$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="b">b条</trans>}}{{\displaystyle <trans\; data-src="\sum ">\sum </trans>}}}{<trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="a">一</trans>}\\ \mathrm{<trans\; data-src="s">秒</trans>}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="\{">\{</trans>\begin{array}{c}\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}\\ \mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}\end{array}<trans\; data-src="\}">\}</trans><trans\; data-src=" "> </trans><trans\; data-src=",">,</trans>$$

(9)

其中花括号表示第二类斯特林数。总之，这让我们猜测方程（5）表示的恒等式，其中方程（9）用于设置$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="k">k个</trans>}$和$\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="j">j个</trans>}$.

递归定义的函数$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$包含变量$\mathrm{<trans\; data-src="u">u个</trans>}$作为$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$。这使它有了一个相当短而优雅的外观，但代价是在更深层次上进行时间密集的递归$\mathrm{<trans\; data-src="r">第页</trans>}$.现在，$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$已通过将递归扩展到特定级别进行变换，从而识别变量幂系数中的模式$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="u">u个</trans>}$，它在代数上看起来更对称，因为名数和分母都随$\mathrm{<trans\; data-src="u">u个</trans>}$。将球分配到盒子中的问题的关联是非常出乎意料的。

3.有效性测试

验证方程式（5）适用于整数的Mathematica代码$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="30">30</trans>}$中提供了图3。前两行定义${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="if">如果</trans>}<trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\notin ">\notin </trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$和${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$分别为。第三行表示${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$如方程式（4）所示。第四行定义了函数$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans><trans\; data-src=" "> </trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans>$类似于方程式（9）。第五行涉及等式（5）左右两侧的差值的计算，逐步通过$\mathrm{<trans\; data-src="r">第页</trans>}$从0到30，增加了更高递归级别所需的执行时间。请注意，此处右侧是通过使用方程式（9）表示的。输出如所需，是一个由零组成的数组，表明对于每个考虑的值$\mathrm{<trans\; data-src="r">第页</trans>}$，推测的恒等式成立。

4.结束语

由于方程（5）两边展开形式的复杂性$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$，毫无疑问，等式（5）一般适用于$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$话虽如此，我们仍将严格证明方程（5）作为一个悬而未决的问题。接近证明的潜在有用参考包括，例如[8,9]，但其中没有一个尝试专门为${\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}$方程（6）的系数或${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}}$方程（7）的系数。在中提供[10]，以及[2]，是无穷和的闭式解，在少数情况下也是负值$\mathrm{<trans\; data-src="r">第页</trans>}$从这些结果中，我们还没有确定出允许我们推测整数方程（1）左侧和的一般闭合形式解的模式$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$.

我们还研究了$\mathrm{<trans\; data-src="u">u个</trans>}$方程（5）的任一侧作为方程（1）无穷和的解$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$级数的数值计算，即方程（1）的LHS的数值计算存在一些困难，因为每个和都包含一个随$\mathrm{<trans\; data-src="k">k个</trans>}$、运行指标和阻尼组件$\mathrm{<trans\; data-src="u">u个</trans>}\mathrm{<trans\; data-src="exp">经验</trans>}<trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$二者结合在一起会在一定范围内产生较大的振幅$\mathrm{<trans\; data-src="k">k个</trans>}$，直到指数阻尼接管。对于负片$\mathrm{<trans\; data-src="u">u个</trans>}$，系列术语交替出现，并且总和中的大数字被取消。几乎在收敛区域的任何地方，都需要高精度的算法。在奇数间隔的$\mathrm{<trans\; data-src="k">k个</trans>}$为负值$\mathrm{<trans\; data-src="u">u个</trans>}$对于非交替情况($\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$)，没有取消连续$\mathrm{<trans\; data-src="k">k个</trans>}$和，因此，和的值变得非常大。该级数在$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$.为实现更高数值的可靠数值评估$\mathrm{<trans\; data-src="r">第页</trans>}$在极点附近，由于涉及的数字较多，这变得更加困难，因此应用以下程序：因为该区域中的总和的值由出现在更高的值时的最大总和决定$\mathrm{<trans\; data-src="k">k个</trans>}$值仅为高$\mathrm{<trans\; data-src="k">k个</trans>}$通过使用$\mathrm{<trans\; data-src="k">k个</trans>}$summand作为$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans>$无穷。这也揭示了复数收敛区域的细节$\mathrm{<trans\; data-src="u">u个</trans>}$，稍后将讨论。正如Mathematica中的计算所证明的那样$\mathrm{<trans\; data-src="u">u个</trans>}$方程（1）的LHS值在极点附近，例如$\mathrm{<trans\; data-src="I">我</trans>}$，通过替换索引上的总和来近似计算$\mathrm{<trans\; data-src="k">k个</trans>}$通过积分：

$$\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="=">=</trans>\frac{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="x">x个</trans>}}}{\sqrt{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\pi ">\pi </trans>}}}{{\displaystyle <trans\; data-src="\int ">\int </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="k">k个</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="exp">经验</trans>}<trans\; data-src="[">[</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="ln">自然对数</trans>}\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="ln">自然对数</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="u">u个</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=" "> </trans><trans\; data-src=".">.</trans>$$

(10)

然后我们可以检查整个范围。的左侧面板图4显示了等式（1）和（5）的对数绝对值作为函数的比较$\mathrm{<trans\; data-src="u">u个</trans>}$特别适用于以下情况$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1.9">1.9</trans>}$和$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$黑色虚线表示方程式（5）的RHS，橙色虚线通过方程式（5的LHS显示结果，蓝色圆点表示序列值。对于后者的计算，积分近似用于$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0.3">0.3</trans>}$总体上一致性良好。

大值的渐近项的行为$\mathrm{<trans\; data-src="k">k个</trans>}$由因素决定$\mathrm{<trans\; data-src="exp">经验</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="ln">自然对数</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="u">u个</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$可以避免爆炸$\mathrm{<trans\; data-src="k">k个</trans>}$只有当$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="ln">自然对数</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="u">u个</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$.对于复杂$\mathrm{<trans\; data-src="u">u个</trans>}$，这将产生一个对称的收敛区域，如右面板所示图4。对于的值$\mathrm{<trans\; data-src="u">u个</trans>}$在这个区域之外，序列会发生分歧。简而言之，我们的评估表明方程（1）适用于非负整数$\mathrm{<trans\; data-src="r">第页</trans>}$和while$\mathrm{<trans\; data-src="u">u个</trans>}$满足以下两个关系：

$$<trans\; data-src="\{">\{</trans>\begin{array}{c}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.27846454">0.27846454</trans>}<trans\; data-src="\dots ">\dots </trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="Re">重新</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}\\ {<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="Im">伊姆河</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="exp">经验</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Re">重新</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="-">-</trans>{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="Re">重新</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}\end{array}<trans\; data-src=";">;</trans>$$

(11)

哪里$\mathrm{<trans\; data-src="Re">重新</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="Im">伊姆河</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>$表示的实部和虚部$\mathrm{<trans\; data-src="u">u个</trans>}$，其中使用Lambert$\mathrm{<trans\; data-src="W">W公司</trans>}$函数实数部分的范围$\mathrm{<trans\; data-src="u">u个</trans>}$以某种对称的方式从$<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans>$到$\mathrm{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans>$，c.f.，方程式（16.6）in[2].