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%I #26 May 20 2019 09:04:20
%S 1,0,5,0,238,0,51508,0,35028576,0,59053389408,0,209726098354368,0,
%T 1397532391623302400,0,16043549794523492290560,0,
%U 297285345537576037788672000,0,8447414796960536731803240038400
%N a(n) = (i^n)*Sum_{k=0..n} (k+1)*B_k*|s(n+2,k+2)|*(n+2)^k.
%C |s(n,k)| is the unsigned Stirling number of first kind (see A008275), B_k is the Bernoulli number and i^2=-1. All even-indexed terms are positive integers, and the odd-indexed terms are zero. A generating function would be welcomed.
%H R. Gy, <a href="http://arxiv.org/abs/1902.09309">An aerated triangular array of integers</a>, arXiv: 1902.09309 [math.CO], 2019.
%t list = {};
%t nlim = 20; Do[s=(-1)^(n/2) Sum[(-1)^(n-k)*(k+1)*BernoulliB[k]*StirlingS1[n+2,k+2]*(n+2)^k,{k,0,n}];AppendTo[list,s], {n,0,nlim}]; Print[list]
%o (PARI) a(n) = (I^n)*sum(k=0, n, (k+1)*bernfrac(k)*abs(stirling(n+2,k+2,1))*(n+2)^k); \\ _Michel Marcus_, May 19 2019
%K nonn
%O 0,3
%A _René Gy_, May 10 2017
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