How to solve this recurrence involving binomial coefficients?
Let us guess that $A(n,k) = \alpha \binom{n}{k} + \beta$. The recurrence reads
$$
\alpha \binom{n}{k} + \beta = \alpha \binom{n-1}{k} + \beta + \alpha \binom{n-1}{k-1} + \beta + c,
$$
from which we deduce $\beta = -c$. The base cases are
$$
\alpha \binom{n}{n} - c = \alpha \binom{n}{0} - c = 1,
$$
From which we deduce $\alpha = c+1$. All in all, the solution is
$$
A(n,k) = (c+1) \binom{n}{k} - c.
$$