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A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).

%I A299106 #11 Mar 31 2018 12:02:08
%S A299106 1,1,2,4,9,19,41,88,189,405,869,1864,3998,8575,18392,39448,84610,
%T A299106 181475,389235,834848,1790617,3840591,8237462,17668057,37895195,
%U A299106 81279216,174331098,373912708,801983781,1720128713,3689404772,7913191304,16972547194,36403436640
%N A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).
%H A299106 Vaclav Kotesovec, <a href="/A299106/b299106.txt">Table of n, a(n) for n = 0..3000</a>
%H A299106 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A299106 G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)).
%F A299106 a(0) = 1; a(n) = Sum_{k=1..n} A000009(k-1)*a(n-k).
%F A299106 a(n) ~ c * d^n, where d = 2.14484226934608840026733598736202689102117985119507858808036465196716739... is the root of the equation QPochhammer(1/d, 1/d^2)*d = 1 and c = 0.4217892515709863296976217395517853732959704351198250451894928058439... = 2/(2+Derivative[0, 1][QPochhammer][-1, 1/d]/d^2). - _Vaclav Kotesovec_, Feb 03 2018, updated Mar 31 2018
%t A299106 nmax = 33; CoefficientList[Series[1/(1 - x Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A299106 nmax = 33; CoefficientList[Series[1/(1 - x/QPochhammer[x, x^2]), {x, 0, nmax}], x]
%t A299106 a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]
%Y A299106 Antidiagonal sums of A286335.
%Y A299106 Cf. A000009, A067687, A299105, A299108.
%K A299106 nonn
%O A299106 0,3
%A A299106 _Ilya Gutkovskiy_, Feb 02 2018