#来自在线整数序列百科全书的问候!http://oeis.org/搜索:id:a189981显示第1-1页,共1页%一A189981%S A189981 1,1,2,12120160028500621138160177924804747216390969920,%电话:A189981 62678679228026584872779520123852417550960862873918454756864,%美国A189981 345553767555348240020444939382463948800012958008875933613962880%N A189981 E.g.f.满足:A(x)=Sum{N>=0}log(1+x*A(x)^N)^N/N!。%C A189981 e.g.f.A(x)的定义是身份的应用:%{a1n*n*0*n=0和{n>=0}二项式(q^n,n)*x^n,q=A(x)。%C A189981考虑函数G(x),使G(x)=1+x*G(x)^p,则%C A189981*G(x)=和{n>=0}log(1+x*G(x)^p)^n/n!(微不足道),以及%C A189981*G(x)=和{n>=0}二项式(p*n+1,n)*x^n/(p*n+1)表示固定p;%C A189981是否存在该序列的e.g.f.的类似表达式?%C A189981注意,术语a(70)-a(83)为负数_Vaclav Kotesovec,2014年7月13日%H A189981瓦茨拉夫·科特索维奇,n=0的n,a(n)表。。106%F A189981 E.g.F.还满足:%F A189981(1)A(x)=和{n>=0}二项式(A(x)^n,n)*x^n。%F A189981(2)A(x)=和{n>=0}x^n*Sum{k=0..n}Stirling1(n,k)*A(x)^(n*k)/n!。%e A189981示例:A(x)=1+x+2*x^2/2!+12*x^3/3!+120*x^4/4!+1600*x^5/5!+。。。%e A189981,其中A(x)满足:%e A189981 A(x)=1+对数(1+x*A(x))+对数(1+x*A(x)^2)^2/2!+日志(1+x*A(x)^3)^3/3!+。。。%e A189981 e.g.f.还满足:%e A189981 A(x)=1+A(x)*x+A(x)^2*(A(x)^2-1)*x^2/2!+A(x)^3*(A(x)^3-1)*(A(x)^3-2)*x^3/3!+A(x)^4*(A(x)^4-1)*(A(x)^4-2)*(A(x)^4-3)*x^4/4!+…+二项式(A(x)^n,n)*x^n+。。。%o A189981(PARI){a(n)=局部(a=1+x);对于(i=1,n,a=sum(m=0,n,log(1+x*(a+x*o(x^n))^m)^m/m!);n!*polcoeff(a,n)}%o A189981(PARI){a(n)=局部(a=1+x);对于(i=1,n,a=和(m=0,n,二项式((a+x*o(x^n))^m,m)*x^m));n!*polcoeff(a,n)}%o A189981(PARI){Stirling1(n,k)=n!*polcoeff(二项式(x,n),k)}%o A189981{a(n)=局部(a=1+x);对于(i=1,n,a=sum(m=0,n,sum(k=0,m,Stirling1(m,k)*(a+x*o(x^n))^(m*k))*x^m/m!);n!*polcoeff(a,n)}%Y A189981,参见A014070、A221101、A224797。%K A189981标志%O A189981 0,3%A A189981保罗·D·汉纳,2011年5月3日#根据OEIS最终用户许可协议提供内容:http://oeis.org/LICENSE