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%I #5 Dec 16 2019 14:25:31
%S 1,2,4,24,136,480,4768,40320,249856,4112640,39563008,319334400,
%T 6249389056,82473431040,1044235737088,20922789888000,355897293438976,
%U 4408265775513600,121616011523719168,2757288942600192000,31308290669925892096
%N Expansion of e.g.f. Sum_{k>=1} tanh(x^k).
%F E.g.f.: Sum_{k>=1} (exp(2*x^k) - 1) / (exp(2*x^k) + 1).
%F a(n) = n! * Sum_{d|n} A155585(d) / d!.
%F a(n) = n! * Sum_{d|n, d odd} 2^(d + 1) * (2^(d + 1) - 1) * Bernoulli(d + 1) / (d + 1)!.
%t nmax = 21; CoefficientList[Series[Sum[Tanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t 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}]
%t Table[n! DivisorSum[n, 2^(# + 1) (2^(# + 1) - 1) BernoulliB[# + 1]/(# + 1)! &, OddQ[#] &], {n, 1, 21}]
%Y Cf. A000182, A009006, A155585, A176475, A330254, A330255, A330505.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Dec 16 2019
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