I have a sum $$ \sum_{k=1}^T(-1)^{T-k}(k!)^2{n\brace k}\binom{n-k}{T-k}. $$ Is it possible to simplify it? Ideally Stirling numbers should disappear.
EDIT: I was asked about context. This is from a problem, where one should find a coefficient at $x^T$ obtained after summing polynomials $c_kk!x^k(1-x)^{n-k}$, where $c_k$ appear to be the sequence http://oeis.org/A019538 or $k!{n\brace k}$. I wouldn't want to say more about where these polynomials come from, cause this is from a programming contest. Not ongoing - it's from the last year's - but still... The formula checks out for small values of $T$ and $n$, but I need it for large ones.