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%I #11 Jun 14 2018 08:16:41
%S 1,1,4,19,130,1120,11960,151595,2230550,37361755,701873371,
%T 14610774346,333746628499,8298025724194,223049950124065,
%U 6444634486214748,199165237980655863,6555102341516877027,228905611339161301812,8452656930719845696590,329075775511339959533232,13471099892869946627980017
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k-1)!!.
%C Euler transform of A001147.
%H Seiichi Manyama, <a href="/A305867/b305867.txt">Table of n, a(n) for n = 0..404</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^A001147(k).
%p N:= 25:
%p S:=series(mul((1-x^k)^(-doublefactorial(2*k-1)),k=1..N),x,N+1):
%p seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Jun 12 2018
%t nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
%Y Cf. A001147, A107895, A179327, A261047, A280088, A305868, A305869.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 12 2018
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