I need to find Generating function of :
$$\left\{\left(^9_n\right)\right\}_{n=0}^{\infty }$$
$$\left\{n\left(^9_n\right)\right\}_{n=0}^{\infty }$$
my suggest answer for the first one:
$$A(x)=\sum _{n=0}^{\infty }\:\left(^9_n\right)x^n=\sum _{n=0}^{9}\:\frac{9!}{n!\left(9-n\right)!}+0+0+0+...$$
so we get:
$$A(x)=\sum _{n=0}^9\:\left(^9_n\right)x^n=(1+x)^9$$
is my answer for the first one correct and any idea how to solve the second one ?
the second one :
$$\frac{d}{dx}\left(\sum _{n=0}^9\:\left(^9_n\right)x^n\right)=\sum \:_{n=0}^9n\left(^9_n\right)x^{n-1}=9\left(1+x\right)^8$$
multiply it with $x$, we get: $$\sum \:_{n=0}^9n\left(^9_n\right)x^n=9x\left(1+x\right)^8$$
so for the second one $$B(x)=9x\left(1+x\right)^8$$
thanks