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%I #21 Oct 19 2023 16:59:08
%S 0,5,75,1000,13125,171875,2250000,29453125,385546875,5046875000,
%T 66064453125,864794921875,11320312500000,148184814453125,
%U 1939764404296875,25391845703125000,332383575439453125,4350957489013671875,56954772949218750000,745547657012939453125
%N a(n) = Sum_{k = 1..2*n} binomial(2*n,k) * Fibonacci(2*k).
%H Reinhard Zumkeller, <a href="/A219462/b219462.txt">Table of n, a(n) for n = 0..500</a>
%H Doron Zeilbeger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enquiry.pdf">An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding</a>, (2007); page 10.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (15,-25).
%F a(n) = Sum_{k=1..n} A034870(n,k)*A001906(k).
%F a(n) = 5^n * Fibonacci(2*n) = A000351(n) * A001906(n).
%F G.f.: 5*x/(25*x^2-15*x+1). - _Colin Barker_, Dec 03 2012
%F E.g.f.: 2*exp(15*x/2)*sinh(5*sqrt(5)*x/2)/sqrt(5). - _Stefano Spezia_, Oct 19 2023
%t Table[Sum[Binomial[2n,k]Fibonacci[2k],{k,2n}],{n,0,20}] (* _Harvey P. Dale_, Aug 26 2017 *)
%o (Haskell)
%o a219462 = sum . zipWith (*) a001906_list . a034870_row
%o (PARI) a(n) = sum(k = 1, 2*n, binomial(2*n,k) * fibonacci(2*k)); \\ _Michel Marcus_, Jan 26 2022
%Y Cf. A000045, A000351, A001906, A007318, A034870, A036291, A087453.
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Nov 20 2012
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