ordinary generating function of some sequence

Question

What is the ordinary generating function of the sequence whose general term is $a_n = {n+k \choose k}$?. I cannot find it in the list given in the book generatingfunctionology, by Herbert S. Wilf. Is there a more comprehensive list?

Answer 1

Try $\displaystyle \frac{1}{(1-x)^{k+1}}$. Recall that $\binom{n}{k} = \binom{n}{n-k}$,and this is 2.5.7 on page 53 of generatingfunctionology.

Answer 2

$a_n=\binom{n+k}{k}=\binom{n+k}{n}$, aslo $$(1+x)^{k+n}=\sum_{i=1}^{\infty}\binom{i+k}{i}x^i$$, so generating function of the sequence is $(1+x)^{k+n}$