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%I #6 Dec 16 2019 14:25:37
%S 1,2,8,24,144,960,5760,40320,524160,4354560,43545600,638668800,
%T 6706022400,99632332800,2092278988800,20922789888000,376610217984000,
%U 9247873130496000,128047474114560000,2919482409811968000,77852864261652480000
%N Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).
%F E.g.f.: -log(theta_4(x)) / 2.
%F E.g.f.: (1/2) * Sum_{k>=1} log((1 + x^k) / (1 - x^k)).
%F E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)).
%F E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1) * (1 - x^(2*k - 1))).
%F exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128.
%F a(n) = (n - 1)! * Sum_{d|n, n/d odd} d.
%t nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]
%Y Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Dec 16 2019
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