Let $$ a_n(i,j)=\sum_{k=\max(i,j)}^{n}s(n,k) S(k,i) S(k,j), $$ here $s, S$ are the Stirling numbers of the first and second kind. I want to simplify the expression. So far I have got the following \begin{align*} a_n(n,i)&=S(n,i),\\ a_n(n-1,i)&=\binom{n}{2}\big(S(n,i)-S(n-1,i)\big)\\ a_n(n-2,i)&=\frac{1}{4}\binom{n}{3}\big((3n-5)S(n,i)-6(n-1)S(n-1,i)+(3n-1)S(n-2,i)\big) \end{align*}
Is there a closed expression for the sum?