标签：infty frac 函数 limits 32 sum 生成 应用 binom

生成函数应用

1. 二项式系数上指标扩展至整数域

\[\binom n m=\binom {m-n-1} m \]

2. 常见生成函数

\(G(x)\)	\(a_k\)
\(\frac 1 {1-x}=\sum\limits_{k=0}^\infty x^k\)	1
\(\frac 1 {1-ax}=\sum\limits_{k=0}^\infty a^kx^k\)	\(a^k\)
\(\frac 1 {(1-x)^2}=\sum\limits_{k=0}^\infty (k+1)x^k\)	\(k+1\)

3.应用

\(\sum\limits_{k=1}^6a_k=50,(a_i\geq3,a_i\in Z)\)，求解集个数。

\[\begin{aligned} \\&[x^{50}](\sum\limits_{k=3}^\infty x^k)^6 \\&=[x^{50}](\frac {x^{3}} {1-x})^6 \\&=[x^{32}](1-x)^{-6} \\&=\binom {-6} {32} \\&=\binom {37} {32} \end{aligned} \]

标签：infty,frac,函数,limits,32,sum,生成,应用,binom
From： https://www.cnblogs.com/subtlemaple/p/17300607.html