스털링 수

개요

\[(x)_{k}=\sum_{j}s(k,j)x^{j}\]

\[x^{k}=\sum_{j}S(k,j)(x)_j\]

\[(x)_{k}=\sum_{j}s(k,j)x^{j}\]

\[(x)_3=x(x-1)(x-2)=2x-3x^2+x^3\] \[s(3,0)=0, s(3,1)=2,s(3,2)=-3,s(3,3)=1\]

\[ s(n,k)=s(n-1,k-1)-(n-1)s(n-1,k) \]

\[x^{n}=\sum_{j}S(n,j)(x)_j\]

\[x^3 = (x)_1+3(x)_2+(x)_3=x+3x(x-1)+x(x-1)(x-2)\] \[S(3,0)=0, S(3,1)=1,S(3,2)=3,s(3,3)=1\]

\[ S(n,k)=S(n-1,k-1)+kS(n-1,k) \]

\[\sum_{k}S(k,n)x^k=\frac{x^n}{(1-x)(1-2x)\cdots(1-nx)}\]

\[\sum_{k}\frac{S(k,n)}{k!}x^k=\frac{(e^x-1)^{n}}{n!}\]

\[\sum_{n=0}^\infty \frac{B_n}{n!} x^n = e^{e^x-1}.\]

Zhao, Wei, Jianrong Zhao, and Shaofang Hong. “The 2-Adic Valuations of Differences of Stirling Numbers of the Second Kind.” arXiv:1407.8443 [math], July 31, 2014. http://arxiv.org/abs/1407.8443.