A261330 Euler transform of Pell-Lucas numbers.

1, 2, 9, 30, 106, 348, 1153, 3698, 11798, 37034, 115294, 355202, 1086080, 3294912, 9931019, 29745296, 88597104, 262508288, 774073787, 2272321666, 6642701371, 19342768210, 56117550874, 162247236638, 467563212923, 1343273262184, 3847866714452, 10991864363660

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. A002203, A261329, A261331, A261332, A166861, A261031.

Programs

Mathematica

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

Formula

G.f.: Product_{k>=1} 1/(1-x^k)^(A002203(k)).

a(n) ~ (1+sqrt(2))^n * exp(-1 + 2^(-3/2) + 2*sqrt(n) + s) / (2 * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} = 2/(((1+sqrt(2))^k + 2/(1 + (1+sqrt(2))^k) - 3)*k) = 0.40371233206538058741995064489690066306587648488344483...