Pensieve header: Multiplication term count.

s[0,_]=1;
s[_,0]=1;
s[m_,n_] := s[m,n] =  (∑_(j=0)^m)Binomial[m+1,j+1]s[m-j,n-1];

Table[s[m,n], {m,0,8}, {n,0,8}] // MatrixForm

Table[s[n,n], {n,0,12}]

Seen at http://oeis.org/A092552

s[12,12]/24! // N