A330511 Expansion of e.g.f. Sum_{k>=1} arctan(x^k).

1, 2, 4, 24, 144, 480, 4320, 40320, 282240, 4354560, 36288000, 319334400, 6706022400, 74724249600, 1046139494400, 20922789888000, 376610217984000, 4979623993344000, 115242726703104000, 2919482409811968000, 29194824098119680000

Offset: 1

Author

Ilya Gutkovskiy, Dec 16 2019

Keywords

• nonn

Crossrefs

Cf. A005359, A050469, A176475, A330504, A330505.

Programs

Mathematica

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

PARI

a(n) = (n-1)!*sumdiv(n, d, if (n/d % 2, (-1)^((n/d - 1)/2)*d)); \\ Michel Marcus, Dec 17 2019

Formula

E.g.f.: Sum_{i>=1} Sum_{j>=1} (-1)^(j + 1) * x^(i*(2*j - 1)) / (2*j - 1).

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