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%I #12 May 06 2022 20:33:37
%S 1,0,1,3,14,80,544,4284,38310,383256,4239006,51345690,675770028,
%T 9600349824,146396925648,2384700728760,41320373582652,758780222426592,
%U 14718569154071964,300706641183038292,6453691377726073128,145154958710291611200,3414131149418742544320
%N Expansion of e.g.f. exp( log(1 - x)^2 / 2 ).
%H Seiichi Manyama, <a href="/A347001/b347001.txt">Table of n, a(n) for n = 0..448</a>
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,2)| * a(n-k).
%F a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(2^k * k!). - _Seiichi Manyama_, May 06 2022
%t nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/(2^k*k!)); \\ _Seiichi Manyama_, May 06 2022
%Y Cf. A000254, A060311, A305306, A346921, A347002, A347003, A347004.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Aug 10 2021
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