生成函数求问

令 $S=f(x)$，则 $xS=\sum_{i=0}^{\infty}\limits i^2x^{i+1}=\sum_{i=0}^{\infty}\limits ((i+1)^2-2i-1)x^{i+1}=\sum_{i=0}^{\infty}\limits(i+1)^2x^{i+1}-2\sum_{i=0}^{\infty}\limits(i+1)x^{i+1}+\sum_{i=0}^{\infty}\limits x^{i+1}=S+\frac{x}{1-x}-\frac{2}{(1-2x)^2}$，从而$S=\frac{\frac{x}{1-x}-\frac{2}{(1-2x)^2}}{x-1}$。