A261331 Expansion of Product_{k>=1} (1+x^k)^(A000129(k)).

1, 1, 2, 7, 18, 52, 143, 396, 1083, 2971, 8087, 21981, 59533, 160857, 433467, 1165542, 3126951, 8372451, 22374172, 59684669, 158941356, 422582925, 1121814072, 2973703449, 7871754065, 20809918535, 54943916547, 144891525408, 381647503607, 1004149670985

Offset: 0

Author

Vaclav Kotesovec, Aug 15 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
• Eric Weisstein's World of Mathematics, Pell Number
• Wikipedia, Pell number

Crossrefs

Cf. A000129, A261329, A261330, A261332, A261050, A261051.

Programs

Mathematica

nmax=40; Pell[0]=0; Pell[1]=1; Pell[n_]:=Pell[n] = 2*Pell[n-1] + Pell[n-2]; CoefficientList[Series[Product[(1+x^k)^Pell[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (1+sqrt(2))^n * exp(-1/8 + 2^(1/4)*sqrt(n) + s) / (2^(11/8) * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} (-1)^(k+1)/(((sqrt(2)+1)^k - (sqrt(2)-1)^k - 2)*k) = -0.1149083344289588668149210160138124159112948627968378825745674888...

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - 2*x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018