A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).

1, 1, 3, 5, 11, 18, 36, 59, 109, 181, 318, 525, 902, 1481, 2492, 4087, 6788, 11090, 18274, 29776, 48772, 79332, 129411, 210172, 341958, 554728, 900872, 1460298, 2368555, 3837147, 6218652, 10070389, 16311432, 26407350, 42757335, 69208746, 112032256, 181316714

Offset: 0

Author

Franklin T. Adams-Watters, Jul 16 2009

Keywords

• nonn
• easy

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000045, A000041, A001156, A003105, A263401, A276527.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(&*[(1-x^k-x^(2*k)): k in [1..100]]))); // G. C. Greubel, Oct 24 2018

Maple

F:= n-> combinat[fibonacci](n+1):
b:= proc(n, i) option remember; `if`(n=0 or i=1, F(n),
      add((t-> b(t, min(t, i-1)))(n-i*j)*F(j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..39);  # Alois P. Heinz, Aug 24 2019

Mathematica

nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2016 *)

PARI

al(n)=Vec(1/prod(k=1,n,1-x^k-x^(2*k)+x*O(x^n)))

Formula

a(n) ~ p / (sqrt(5) * r^(n+1)), where r = (sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 - r^n - r^(2*n)) = 4.64451592505133910330213147... . - Vaclav Kotesovec, Nov 16 2016