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%I #37 Apr 01 2021 22:21:43
%S 1,1,4,17,104,661,5584,47837,483584,5332681,63940864,802442057,
%T 11548580864,170258934301,2602357970944,44379608478677,
%U 800966933970944,14221966162901521,277738909303373824,5823354583392253697,121050262784565837824,2668717158207399650341,62376912442894992277504
%N Expansion of e.g.f. Product_{k>0} 1/(1-sin(x)^k).
%C a(46) is negative. - _Vaclav Kotesovec_, Oct 03 2020
%H Seiichi Manyama, <a href="/A335626/b335626.txt">Table of n, a(n) for n = 0..400</a> (terms n = 0..100 from Vaclav Kotesovec)
%F E.g.f.: exp( Sum_{k>0} sigma(k)*sin(x)^k/k ).
%t nmax = 25; CoefficientList[Series[Product[1/(1 - Sin[x]^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Oct 03 2020 *)
%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/eta(sin(x))))
%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-sin(x)^k)))
%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k)*sin(x)^k/k))))
%Y Cf. A000041, A335627, A335629.
%K sign
%O 0,3
%A _Seiichi Manyama_, Oct 02 2020