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

%I A162891 #24 Sep 08 2022 08:45:46
%S A162891 1,1,3,5,11,18,36,59,109,181,318,525,902,1481,2492,4087,6788,11090,
%T A162891 18274,29776,48772,79332,129411,210172,341958,554728,900872,1460298,
%U A162891 2368555,3837147,6218652,10070389,16311432,26407350,42757335,69208746,112032256,181316714
%N A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).
%H A162891 Vaclav Kotesovec, <a href="/A162891/b162891.txt">Table of n, a(n) for n = 0..1000</a>
%F A162891 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
%p A162891 F:= n-> combinat[fibonacci](n+1):
%p A162891 b:= proc(n, i) option remember; `if`(n=0 or i=1, F(n),
%p A162891       add((t-> b(t, min(t, i-1)))(n-i*j)*F(j), j=0..n/i))
%p A162891     end:
%p A162891 a:= n-> b(n$2):
%p A162891 seq(a(n), n=0..39);  # _Alois P. Heinz_, Aug 24 2019
%t A162891 nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 16 2016 *)
%o A162891 (PARI) al(n)=Vec(1/prod(k=1,n,1-x^k-x^(2*k)+x*O(x^n)))
%o A162891 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(&*[(1-x^k-x^(2*k)): k in [1..100]]))); // _G. C. Greubel_, Oct 24 2018
%Y A162891 Cf. A000045, A000041, A001156, A003105, A263401, A276527.
%K A162891 nonn,easy
%O A162891 0,3
%A A162891 _Franklin T. Adams-Watters_, Jul 16 2009