Expressing one generating function like combination of another generating functions.

Let A (t), B (t) and C (t) - generating functions for sequences $a_0, a_1, a_2,\dots; b_0, b_1, b_2,\dots and\ c_0, c_1, c_2,\dots$ Express C (t) through A (t) and B (t), if $c_n=\sum_{j+4k<=n}a_jb_k$ . I tried to build series for c_n and then express from it series for a_n and b_n but without success. Please help me. Thank you in advance.

If it's not easy to see the connection between $C(x)=\sum_{n=0}^{\infty}c_nx^n$ with $A(x)=\sum_{j=0}^{\infty}a_jx^j$ and $B(x)=\sum_{k=0}^{\infty}c_kx^k$ we could look at the coefficients $c_n$ for small $n$ and check if we detect some pattern or regularity.

Assuming $c_n$ is given by \begin{align*} c_n=\sum_{{j+4k=n}\atop{j,k\geq 0}}a_jb_k\qquad n\geq 0 \end{align*}

we see:

We observe the index of $b_n$ is increasing by one whenever the index of $c_n$ is increased by $4$. So let's start with:
Multiplication of $A(x)$ and $B(x^4)$ results in: