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%I #7 Jan 27 2020 02:40:19
%S 1,1,3,14,85,632,5559,56352,645929,8252352,116189291,1786361216,
%T 29764770941,534082233856,10264484355103,210312181051392,
%U 4575364233983057,105310034714202112,2556360647841415379,65261358332774277120,1747713179543456515749
%N E.g.f.: exp(1 / (1 - sinh(x)) - 1).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A006154(k) * a(n-k).
%F a(n) ~ exp(1/(2^(3/2) * log(1 + sqrt(2))) - 3/4 + 2^(3/4) * sqrt(n) / sqrt(log(1 + sqrt(2))) - n) * n^(n - 1/4) / (2^(5/8) * log(1 + sqrt(2))^(n + 1/4)). - _Vaclav Kotesovec_, Jan 27 2020
%t nmax = 20; CoefficientList[Series[Exp[1/(1 - Sinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A006154[n_] := Sum[Sum[(-1)^j (k - 2 j)^n Binomial[k, j]/2^k, {j, 0, k}], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A006154[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%Y Cf. A003704, A003724, A006154, A075729, A331607, A331611.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 22 2020