generating function small question about coefficient

Question

generating function of a0+a1... C((n+4),4)

can anyone help?

small step in proof

thank you so much

please show steps

Answer 1

If what you are seeking is a closed form for $$ f(x)=\sum_{n=0}^\infty\binom{n+4}{4}x^n\tag{1} $$ then consider negative binomial coefficients. $(1)$ becomes $$ \begin{align} f(x) &=\sum_{n=0}^\infty\binom{n+4}{4}x^n\\ &=\sum_{n=0}^\infty\binom{n+4}{n}x^n\\ &=\sum_{n=0}^\infty\binom{-5}{n}(-1)^nx^n\\[6pt] &=(1-x)^{-5}\tag{2} \end{align} $$