Q:

On the number of monic polynomials

Let $a,b,c,d\in\Bbb N$ with $c<b$.
Let $N_+(a,b,c,d)$ be the number of monic polynomials $f\in \Bbb Z[x]$of degree $d$ with non-negative coefficients such that $$f(a)=b$$ $$f(0)=c$$
What is the value of $$\sum_{d=0}^tN_+(a,b,c,d)?$$
Are there sharp estimates?

A:

Not a complete answer, but note that by definition $N_+(a,b,c,d)$ is the number of ways of writing $b$ as a sum $b=\sum_{k=0}^d c_ka^k$ with non-negative $c_k$, and with  $c_0=c$ and $c_d=1$. In particular it is certainly zero unless $a|(b-c)$ and $b-c\ge a^d$, in which case $N_+(a,b,c,d):=A\big(\frac{b-c-a^d}{a}\big)$, where $A(n)$ is the number of ways of writing $n$ as a sum $n=\sum_{k=0}^{d-2}x_k a^k$, with non-negative integers $x_k$, therefore corresponding to a g.f.
$$\sum_{n=0}^\infty A(n)x^n=\frac{1}{(1-x)(1-x^a)(1-x^{a^2})\dots (1-x^{a^{d-2}})   }\ .$$

