Generating-function problem

Question

$A(t),B(t) $ and $ C(t)$ are generating functions for sequences $a_0,a_1,a_2,...;b_0,b_1,b_2,...;c_0,c_1,c_2,\dots$ I do not know how to express C(t) through A(t) and B(t), if $c(n)=\sum_{k = 0}^{[n / 4]} a_k b_{n-4k}$

Answer 1

First one may reduce the generating function of B(t) to the generating function which keeps only each 4th term of b, say Bmod(t). Then c is the convolution product of the two sequences and therefore the generating function C(t) the product of the two generating functions A(t)*Bmod(t).

Generating Bmod(t) could be done in 2 steps of dissecting the sequence b, because (B(t)+B(-t))/2 keeps only the b(0), b(2) etc at even indices, and (B(t)-B(-t))/2 only the b at odd indices.