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

1,2

Links

Table of n, a(n) for 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!.

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}]

Crossrefs

Cf. A001818, A330254, A330505, A330506.

Sequence in context: A176473 A185056 A346204 * A194349 A208969 A020022

Adjacent sequences: A330509 A330510 A330511 * A330513 A330514 A330515

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 16 2019

Status

approved