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例子
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考虑满足以下条件的幂级数G(x,n)族:
x=和{m>=1}1/G(x,n)^(n*m)*乘积{k=1..m}(1-1/G(x,n)*k)。
带有g.f.g(x,n)的序列示例如下:
n=2:A001002号= [1, 1, 1, 3, 10, 38, 154, 654, 2871, 12925, ...];
n=3:1997年11月= [1, 1, 2, 9, 46, 259, 1539, 9484, 59961, ...];
n=4:A181998号= [1, 1, 3, 18, 124, 935, 7443, 61510, 522467, ...];
n=5:A209441型= [1, 1, 4, 30, 260, 2463, 24656, 256493, 2745149, ...];
n=6:A209442型= [1, 1, 5, 45, 470, 5365, 64766, 813012, 10505163, ...]; ...
观察序列反转(G(x,n)-1)由多项式给出:
n=1:x;
n=2:x-x^2-x^3;
n=3:x-2*x^2-x^3+4*x^4+4*x^5+x^6;
n=4:x-3*x ^2+11*x ^4+x ^5-30*x ^6-42*x ^7-26*x ^8-8*x ^9-x ^10;
n=5:x-4*x^2+2*x^3+20*x^4-19*x^5-100*x^6+3*x^7+403*x^8+808*x^9+861*x^10+584*x^11+262*x^12+76*x^13+13*x^14+x^15。。。
上述多项式中的系数三角形开始于:
[1];
[1, -1, -1];
[1, -2, -1, 4, 4, 1];
[1, -3, 0, 11, 1, -30, -42, -26, -8, -1];
[1, -4, 2, 20, -19, -100, 3, 403, 808, 861, 584, 262, 76, 13, 1];
[1, -5, 5, 30, -65, -191, 378, 1557, 103, -8551, -23911, -37958, -41831, -34156, -21179, -10015, -3571, -933, -169, -19, -1];
[1, -6, 9, 40, -145, -261, 1384, 2897, -8980, -38710, -14146, 258401, 990407, 2170834, 3426095, 4198850, 4137440, 3336534, 2220430, 1221799, 554027, 205250, 61206, 14351, 2550, 323, 26, 1];
[1, -7, 14, 49, -266, -245, 3325, 2596, -36710, -70556, 281645, 1413916, 1184890, -10255248, -54012830, -156371880, -329973512, -552895722, -765517470, -895408431, -896614676, -774834055, -580511469, -377792286, -213512611, -104550572, -44163315, -15985147, -4910774, -1263620, -267378, -45321, -5918, -559, -34, -1]; ...
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