A330506 Expansion of e.g.f. Sum_{k>=1} arcsin(x^k).

1, 2, 7, 24, 129, 840, 5265, 40320, 434385, 3900960, 40809825, 558835200, 6335076825, 91070179200, 1641957141825, 20922789888000, 359796258446625, 7663952552256000, 122832552380162625, 2615369658789888000, 62315614994643635625

Offset: 1

Author

Ilya Gutkovskiy, Dec 16 2019

Keywords

• nonn

Crossrefs

Cf. A001818, A176473, A184877, A330254, A330505.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Sum[ArcSin[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! DivisorSum[n, ((# - 2)!!)^2/#! &, OddQ[#] &], {n, 1, 21}]

Formula

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