图沙德多项式

其中${\displaystyle S(n,k)}$是第二类斯特林数

Touchard Polynomials

图沙德多项式是1939年法兰西数学家Jacques Touchard提出的多项式。定义如下[1]

Touchard Polynomials

${\displaystyle T_{0}(x)=1,\qquad T_{n}(x)=\sum _{k=1}^{n}S(n,k)x^{k}=\sum _{k=1}^{n}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}x^{k},\quad n>0,}$

前面几个图沙德多项式是：

• ${\displaystyle T_{0}(x)=1}$
• ${\displaystyle T_{1}(x)=x}$
• ${\displaystyle T_{2}(x)=2*x+2*x^{2}}$
• ${\displaystyle T_{3}(x)=36*x+108*x^{2}+36*x^{3}}$
• ${\displaystyle T_{4}(x)=13824*x+96768*x^{2}+82944*x^{3}+13824*x^{4}}$
• ${\displaystyle T_{5}(x)=207360000*x+3110400000*x^{2}+5184000000*x^{3}+2073600000*x^{4}+207360000*x^{5}}$