%I #4 Dec 17 2019 08:41:02

%S 1,2,5,24,129,600,4815,40320,313425,3900960,39023775,399168000,

%T 6335076825,83286403200,1169542749375,20922789888000,359796258446625,

%U 5529827983680000,120457648437501375,2615369658789888000,40723609672075955625

%N Expansion of e.g.f. Sum_{k>=1} arcsinh(x^k).

%F E.g.f.: Sum_{k>=1} log(x^k + sqrt(1 + x^(2*k))).

%F a(n) = n! * Sum_{d|n, d odd} (-1)^((d - 1)/2) * ((d - 2)!!)^2 / d!.

%t nmax = 21; CoefficientList[Series[Sum[ArcSinh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%t Table[n! DivisorSum[n, (-1)^((# - 1)/2) ((# - 2)!!)^2/#! &, OddQ[#] &], {n, 1, 21}]

%Y Cf. A001818, A330254, A330505, A330506.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Dec 16 2019