A219312 Composition of the binomial transform of Fibonacci numbers and the Catalan transform of Fibonacci numbers.

0, 1, 4, 15, 59, 243, 1034, 4501, 19920, 89281, 404184, 1844789, 8477571, 39183625, 182010366, 849115811, 3976405347, 18684473203, 88060677880, 416162484693, 1971567963673, 9361218368921, 44539107835094, 212308063827055, 1013779444844754, 4848597239921803

Offset: 0

Author

Arkadiusz Wesolowski, Nov 17 2012

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, A Catalan transform and related transformations on integer sequences, pp. 20-22.

Crossrefs

Cf. A000045.

Programs

Mathematica

CoefficientList[Series[(Sqrt[5*x-1] - Sqrt[x-1])/(2*((x-1)*Sqrt[5*x-1] - x*Sqrt[x-1])), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 19 2013 *)

PARI

Vec((sqrt(5*x-1) - sqrt(x-1))/(2*((x-1)*sqrt(5*x-1) - x*sqrt(x-1))) + O(x^25)) \\ G. C. Greubel, Jan 28 2017

Formula

G.f.: (sqrt(5*x-1) - sqrt(x-1))/(2*((x-1)*sqrt(5*x-1) - x*sqrt(x-1))).

a(n) ~ 5^(n+5/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 19 2013

D-finite with recurrence n*a(n) +4*(-3*n+2)*a(n-1) +(45*n-58)*a(n-2) +2*(-27*n+46)*a(n-3) +20*(n-2)*a(n-4)=0. - R. J. Mathar, Nov 22 2024