A260787 G.f.: Product_{k>=1} 1/(1-x^k)^Fibonacci(k+2).

1, 2, 6, 15, 38, 89, 210, 474, 1065, 2339, 5091, 10919, 23230, 48887, 102126, 211599, 435561, 890617, 1810786, 3661118, 7365473, 14747049, 29397160, 58356179, 115392801, 227332038, 446304671, 873298579, 1703463864, 3312873935, 6424553973, 12425158365, 23968214357, 46120280910, 88535346223

Offset: 0

Author

N. J. A. Sloane, Aug 05 2015

Keywords

• nonn

Comments

In general, the sequence with g.f. Product_{k>=1} 1/(1-x^k)^Fibonacci(k+z), where z is nonnegative integer, is asymptotic to phi^(n + z/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp((phi/10 - 1/2) * Fibonacci(z) - Fibonacci(z+1)/10 + 2 * 5^(-1/4) * phi^(z/2) * sqrt(n) + s), where s = Sum_{k>=2} (Fibonacci(z) + Fibonacci(z+1) * phi^k) / ((phi^(2*k) - phi^k - 1)*k) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..4450
• W. S. Gray, K. Ebrahimi-Fard, Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra, arXiv:1411.0222 [math.OC], 2014. See F_H.
• Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
• Vaclav Kotesovec, Asymptotics of sequence A034691

Crossrefs

Cf. A000045, A034691, A166861, A200544.

Programs

Mathematica

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k+2], {k, 1, 20}], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 05 2015 *)

Formula

a(n) ~ phi^(n+1/2) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 7/10 + 2*5^(-1/4)*phi*sqrt(n) + s), where s = Sum_{k>=2} (1 + 2*phi^k) / ((phi^(2*k) - phi^k - 1)*k) = 1.39069800276768443926918973402733105305129194986259856042723... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015