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f[n_] := Sum[Binomial[n-k+1, k] (-1)^k (n-k+1)!, {k, 0, Quotient[n+1, 2]}];
a[n_] := Sum[f[k] f[n-k], {k, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 14 2023 *)
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Alois P. Heinz, <a href="/A227096/b227096.txt">Table of n, a(n) for n = 0..200</a>
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a(n) ~ 2*n*n!/exp(1). - Vaclav Kotesovec, Jul 08 2013
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1, 2, 5, 20, 104, 632, 4396, 34680, 307236, 3026472, 32849364, 389704800, 5017492320, 69678231552, 1038078389376, 16513758904320, 279354776803200, 5007072973075200, 94783054774919040, 1889504358498754560, 39565281716813111040, 868194780280625779200
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