A330512 Expansion of e.g.f. Sum_{k>=1} arcsinh(x^k).
1, 2, 5, 24, 129, 600, 4815, 40320, 313425, 3900960, 39023775, 399168000, 6335076825, 83286403200, 1169542749375, 20922789888000, 359796258446625, 5529827983680000, 120457648437501375, 2615369658789888000, 40723609672075955625
Offset: 1
Keywords
Programs
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Mathematica
nmax = 21; CoefficientList[Series[Sum[ArcSinh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest Table[n! DivisorSum[n, (-1)^((# - 1)/2) ((# - 2)!!)^2/#! &, OddQ[#] &], {n, 1, 21}]
Formula
E.g.f.: Sum_{k>=1} log(x^k + sqrt(1 + x^(2*k))).
a(n) = n! * Sum_{d|n, d odd} (-1)^((d - 1)/2) * ((d - 2)!!)^2 / d!.