Determine generating function for given sequence.

Question

Let $A(x) $ be generating function for sequence $a_n$ and let $s_n = \sum^{n}_{i=0} a_i $. Determine function generating sequence $a_n$

I am asking for an advice. The generating function makes me some problem. Thanks in advance.

Answer 1

You want the notion of the convolution of two sequences. The convolution of the sequences $\alpha=\langle a_n:n\in\Bbb N\rangle$ and $\beta=\langle b_n:n\in\Bbb N\rangle$ is the sequence $\gamma=\langle c_n:n\in\Bbb N\rangle$ such that

$$c_n=\sum_{k=0}^na_kb_{n-k}\;.$$

If $A(x)$ is the generating function for $\alpha$, and $B(x)$ is the generating function for $\beta$, then the generating function for the convolution $\gamma$ is the product $A(x)B(x)$.

For your problem, note that

$$\sum_{k=0}^na_k=\sum_{k=0}^n(a_k\cdot1)\;,$$

so the sequence $\langle s_n:n\in\Bbb N\rangle$ is the convolution of $\langle a_n:n\in\Bbb N\rangle$ and what other sequence $\langle b_n:n\in\Bbb N\rangle$? And what is the generating function of that other sequence?