A219462 a(n) = Sum_{k = 1..2*n} binomial(2*n,k) * Fibonacci(2*k).

0, 5, 75, 1000, 13125, 171875, 2250000, 29453125, 385546875, 5046875000, 66064453125, 864794921875, 11320312500000, 148184814453125, 1939764404296875, 25391845703125000, 332383575439453125, 4350957489013671875, 56954772949218750000, 745547657012939453125

Offset: 0

Author

Reinhard Zumkeller, Nov 20 2012

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..500
• Doron Zeilbeger, An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding, (2007); page 10.
• Index entries for linear recurrences with constant coefficients, signature (15,-25).

Crossrefs

Cf. A000045, A000351, A001906, A007318, A034870, A036291, A087453.

Programs

Haskell

a219462 = sum . zipWith (*) a001906_list . a034870_row

Mathematica

Table[Sum[Binomial[2n,k]Fibonacci[2k],{k,2n}],{n,0,20}] (* Harvey P. Dale, Aug 26 2017 *)

PARI

a(n) = sum(k = 1, 2*n, binomial(2*n,k) * fibonacci(2*k)); \\ Michel Marcus, Jan 26 2022

Formula

a(n) = Sum_{k=1..n} A034870(n,k)*A001906(k).

a(n) = 5^n * Fibonacci(2*n) = A000351(n) * A001906(n).

G.f.: 5*x/(25*x^2-15*x+1). - Colin Barker, Dec 03 2012

E.g.f.: 2*exp(15*x/2)*sinh(5*sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Oct 19 2023