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A305869 Expansion of Product_{k>=1} (1 + x^k)^(2*k-1)!!.

%I A305869 #10 Feb 16 2025 08:33:54
%S A305869 1,1,3,18,123,1098,11806,150406,2218065,37206485,699604235,
%T A305869 14572941915,333037896380,8283300923765,222714069807495,
%U A305869 6436292165450693,198941178161054798,6548632634238445779,228705772883364303114,8446082393596031365629,328846269698068735291665,13462627492562640070346824
%N A305869 Expansion of Product_{k>=1} (1 + x^k)^(2*k-1)!!.
%C A305869 Weigh transform of A001147.
%H A305869 Alois P. Heinz, <a href="/A305869/b305869.txt">Table of n, a(n) for n = 0..404</a>
%H A305869 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305869 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>
%H A305869 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A305869 G.f.: Product_{k>=1} (1 + x^k)^A001147(k).
%p A305869 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A305869       binomial(doublefactorial(2*i-1), j)*b(n-i*j, i-1), j=0..n/i)))
%p A305869     end:
%p A305869 a:= n-> b(n$2):
%p A305869 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jun 13 2018
%t A305869 nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
%t A305869 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
%Y A305869 Cf. A001147, A261052, A305867, A305870.
%K A305869 nonn
%O A305869 0,3
%A A305869 _Ilya Gutkovskiy_, Jun 12 2018