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%I #5 Jun 06 2019 21:59:53
%S 1,-1,3,-11,-47,2111,-18589,-671803,24578625,328227551,-44122149981,
%T -17436709707,129678315723089,-1671362093527073,-621466693739336509,
%U 16245508786340273349,4705502566521408713473,-172007688705762528111809,-54084940569279187064459581
%N a(n) = exp(1) * Sum_{k>=0} (-1)^k*(k*(k + 1))^n/k!.
%F a(n) = Sum_{k=0..n} binomial(n,k)*A000587(2*n-k).
%t Table[Exp[1] Sum[(-1)^k (k (k + 1))^n/k!, {k, 0, Infinity}], {n, 0, 18}]
%t Table[Sum[Binomial[n, k] BellB[2 n - k, -1], {k, 0, n}], {n, 0, 18}]
%Y Cf. A000587, A020556, A090211, A094577.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jun 06 2019