A276527 -id:A276527 - cp's OEIS frontend

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

Franklin T. Adams-Watters, Jul 16 2009

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

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

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

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

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

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

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

A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

Original entry on oeis.org

1, 2, 1, 6, 3, 6, 16, 12, 16, 22, 51, 36, 60, 62, 91, 154, 148, 176, 236, 278, 328, 552, 508, 670, 771, 988, 1068, 1438, 1844, 1998, 2401, 2882, 3300, 4030, 4640, 5406, 7212, 7584, 9072, 10480, 12612, 13964, 17024, 18860, 22545, 27298, 30340, 34372, 41068

Offset: 0

Vaclav Kotesovec, Oct 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A162891, A263401, A276527, A293138.

N:= 100:
P:= mul(1+2*x^m- x^(2*m), m=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,n), n=0..N); # Robert Israel, Oct 01 2017

nmax = 100; CoefficientList[Series[Product[1+2*x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2^(3/2) * sqrt(Pi) * n^(3/4)), where c = Pi^2/6 + log(1+sqrt(2))^2/2 + polylog(2, 3-2*sqrt(2))/2 - 2*polylog(2, sqrt(2)-1) = 1.18805291660775259061867850175092520191179528961165451864292...

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

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A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

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cp's OEIS Frontend

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

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A293182 Expansion of Product_{k>=1} (1 + 2*x^k - x^(2*k)).

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Maple

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A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).