A330504 Expansion of e.g.f. Sum_{k>=1} tanh(x^k).

1, 2, 4, 24, 136, 480, 4768, 40320, 249856, 4112640, 39563008, 319334400, 6249389056, 82473431040, 1044235737088, 20922789888000, 355897293438976, 4408265775513600, 121616011523719168, 2757288942600192000, 31308290669925892096

Offset: 1

Author

Ilya Gutkovskiy, Dec 16 2019

Keywords

• nonn

Crossrefs

Cf. A000182, A009006, A155585, A176475, A330254, A330255, A330505.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Sum[Tanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A155585[n_] := Sum[StirlingS2[n, k] (-2)^(n - k) k!, {k, 0, n}]; a[n_] := n! DivisorSum[n, A155585[#]/#! &]; Table[a[n], {n, 1, 21}]
Table[n! DivisorSum[n, 2^(# + 1) (2^(# + 1) - 1) BernoulliB[# + 1]/(# + 1)! &, OddQ[#] &], {n, 1, 21}]

Formula

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

a(n) = n! * Sum_{d|n} A155585(d) / d!.

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