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%I #7 Aug 09 2021 16:50:34
%S 1,3,4,-15,-134,-357,2374,33645,133186,-1288617,-24887906,-130115895,
%T 1666879306,40612637523,262868197414,-4221449488635,-123802488449774,
%U -952293015617937,18497401668708334,632675912865355425,5622243546094977946,-128799294291220310997
%N E.g.f.: log( 1 + (exp(x) - 1)^2 / 2 ).
%F a(n) = Stirling2(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,2) * k * a(k).
%F a(n) ~ -n! * 2^(n+1) * cos(n*arctan(2*arctan(sqrt(2))/log(3))) / (n * (4*arctan(sqrt(2))^2 + log(3)^2)^(n/2)). - _Vaclav Kotesovec_, Aug 09 2021
%t nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
%t a[n_] := a[n] = StirlingS2[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]
%Y Cf. A000225, A003704, A060311, A346975, A346976, A346977.
%K sign
%O 2,2
%A _Ilya Gutkovskiy_, Aug 09 2021
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