Let $f(m,1) = f(1,n) = 1$ for $m \geq 1, n \geq 1,$ and let $f(m, n) = f(m-1,n) + f(m, n-1) + f(m-1,n-1)$ for $m > 1$ and $n > 1.$ also, let $$s(k) = \sum_{a+b=k} f(a, b), \text{ for } a \geq 1, b \geq 1.$$note: the summation notation means to sum over all positive integers $a, b$ such that $a+b=k.$ given that $$s(k+2) = ps(k+1) + qs(k) \text{ for all } k \geq 2,$$for some constants $p$ and $q$, find $pq.$