A299106 - OEIS

%I #11 Mar 31 2018 12:02:08

%S 1,1,2,4,9,19,41,88,189,405,869,1864,3998,8575,18392,39448,84610,

%T 181475,389235,834848,1790617,3840591,8237462,17668057,37895195,

%U 81279216,174331098,373912708,801983781,1720128713,3689404772,7913191304,16972547194,36403436640

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

%H Vaclav Kotesovec, <a href="/A299106/b299106.txt">Table of n, a(n) for n = 0..3000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)).

%F a(0) = 1; a(n) = Sum_{k=1..n} A000009(k-1)*a(n-k).

%F 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 nmax = 33; CoefficientList[Series[1/(1 - x Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

%t nmax = 33; CoefficientList[Series[1/(1 - x/QPochhammer[x, x^2]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

%Y Antidiagonal sums of A286335.

%Y Cf. A000009, A067687, A299105, A299108.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Feb 02 2018