It's Trivial?

Let $a_n$ be defined such that it is the arithmetic mean of the p numbers before it. Let $a_i=i, 1 \leq i \leq p$ .Let $a_\infty=b_p$ for such a sequence.If

$\sum_{p=0}^{20} b_p {20 \choose p}=a*b^c$

Then find $a+b+c$