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?