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%I #6 Dec 17 2019 08:40:54
%S 1,2,4,24,144,480,4320,40320,282240,4354560,36288000,319334400,
%T 6706022400,74724249600,1046139494400,20922789888000,376610217984000,
%U 4979623993344000,115242726703104000,2919482409811968000,29194824098119680000
%N Expansion of e.g.f. Sum_{k>=1} arctan(x^k).
%F E.g.f.: Sum_{i>=1} Sum_{j>=1} (-1)^(j + 1) * x^(i*(2*j - 1)) / (2*j - 1).
%F a(n) = (n - 1)! * Sum_{d|n, n/d odd} (-1)^((n/d - 1)/2) * d.
%t nmax = 21; CoefficientList[Series[Sum[ArcTan[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t Table[(n - 1)! DivisorSum[n, (-1)^((n/# - 1)/2) # &, OddQ[n/#] &], {n, 1, 21}]
%o (PARI) a(n) = (n-1)!*sumdiv(n, d, if (n/d % 2, (-1)^((n/d - 1)/2)*d)); \\ _Michel Marcus_, Dec 17 2019
%Y Cf. A005359, A050469, A176475, A330504, A330505.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Dec 16 2019
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