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

1, 1, 2, 4, 9, 19, 41, 88, 189, 405, 869, 1864, 3998, 8575, 18392, 39448, 84610, 181475, 389235, 834848, 1790617, 3840591, 8237462, 17668057, 37895195, 81279216, 174331098, 373912708, 801983781, 1720128713, 3689404772, 7913191304, 16972547194, 36403436640

Offset: 0

Author

Ilya Gutkovskiy, Feb 02 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
• N. J. A. Sloane, Transforms

Crossrefs

Antidiagonal sums of A286335.

Cf. A000009, A067687, A299105, A299108.

Programs

Mathematica

nmax = 33; CoefficientList[Series[1/(1 - x Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 - x/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

Formula

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

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

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