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%I #19 Apr 10 2022 01:55:54
%S 1,0,2,2,8,24,100,488,2832,19096,147296,1281392,12422864,132870368,
%T 1554525152,19750621216,270817685568,3986140113792,62686410981696,
%U 1048946532137216,18608550117641728,348854564104019072,6891109834644748032,143058034748452036352
%N Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j + 2 * x).
%H Seiichi Manyama, <a href="/A353252/b353252.txt">Table of n, a(n) for n = 0..451</a>
%F a(n) = Sum_{k=0..floor(n/2)} 2^k * |Stirling1(n-k,k)|.
%t a[n_] := Sum[2^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* _Amiram Eldar_, Apr 09 2022 *)
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+2*x)))
%o (PARI) a(n) = sum(k=0, n\2, 2^k*abs(stirling(n-k, k, 1)));
%Y Cf. A352802, A343579, A353253, A353254.
%Y Cf. A097341.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 08 2022