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I killed this after 48 hours, it did not seem to be getting anywhere. In the best of all possible worlds it would spit out n!. If I put the summation in a function it does compute factorials (somewhat inconveniently)

Mathematica 10 Linux Fedora 20 64-bit

Simplify[Sum[(1/(n - k)!) (Sum[
  Sum[(-1)^(i + j) Binomial[k - 2 j, i] Binomial[k - i - 2 j + 1, 
     j] (n - i - 2 j)!, {i, 0, k - 3 j}], {j, 0, 
   k/3}] + (-1)^(k + 1) Sum[
   Binomial[k - 2 j, k + 1 - 3 j] (n - k + j - 1)!, {j, 
    1, (k + 1)/3}]) , {k, 0, n}]]
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A "derivation" of the n! result can be done using FindSequenceFunction. Start with defining f[n_] := your sum. Then create some data that characterises your function data = Table[{n, f[n]}, {n, 50}], and evaluate fsimple = FindSequenceFunction[data]. Finally, simplify this further for for your use case with FullSimplify[fsimple[n], Assumptions -> n \[Element] Integers && n >= 0]. –  Stephen Luttrell Jul 25 at 8:19

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