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?



On the number of monic polynomials
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.
$$\sum\_{n=0}^\infty A(n)x^n=\frac{1}{(1-x)(1-x^a)(1-x^{a^2})\dots (1-x^{a^{d-2}}) }\ .$$


