A261329 Euler transform of Pell numbers.

1, 1, 3, 8, 23, 62, 175, 477, 1319, 3602, 9851, 26779, 72726, 196724, 531157, 1430144, 3842911, 10303055, 27570786, 73637306, 196333303, 522584286, 1388786089, 3685169795, 9764703347, 25838430572, 68282175170, 180221449469, 475102410065, 1251038486529

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, A261330, A261331, A261332, A166861, A261031.

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/(1-x^k)^Pell[k], {k, 1, nmax}], {x, 0, nmax}], x]

SageMath

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 1)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Formula

G.f.: Product_{k>=1} 1/(1-x^k)^(A000129(k)).

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/(((sqrt(2)+1)^k - (sqrt(2)-1)^k - 2)*k) = 0.17615706029370539578355193664752741450665073523628663099586621933373...

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