A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

1, 2, 8, 24, 144, 960, 5760, 40320, 524160, 4354560, 43545600, 638668800, 6706022400, 99632332800, 2092278988800, 20922789888000, 376610217984000, 9247873130496000, 128047474114560000, 2919482409811968000, 77852864261652480000

Offset: 1

Author

Ilya Gutkovskiy, Dec 16 2019

Keywords

• nonn

Crossrefs

Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]

Formula

E.g.f.: -log(theta_4(x)) / 2.

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

E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)).

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

exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128.

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