cp's OEIS Frontend

A219672 a(n) = Sum_{k=0..n} binomial(n,k)^2*Fibonacci(k).

%I A219672 #21 Feb 16 2025 08:33:18
%S A219672 0,1,5,20,87,405,1924,9225,44625,217528,1066725,5256087,26001000,
%T A219672 129053365,642376709,3205403100,16029187391,80309053285,403040543420,
%U A219672 2025751379997,10195547237235,51376594943136,259180112907875,1308811957775785,6615383878581072
%N A219672 a(n) = Sum_{k=0..n} binomial(n,k)^2*Fibonacci(k).
%H A219672 Vincenzo Librandi, <a href="/A219672/b219672.txt">Table of n, a(n) for n = 0..200</a>.
%H A219672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>.
%F A219672 G.f.: (1/sqrt(1 - (3 + sqrt(5))*x + (3 - sqrt(5))/2*x^2) - 1/sqrt(1 - (3 - sqrt(5))*x + (3 + sqrt(5))/2*x^2))/sqrt(5)
%F A219672 a(n) ~ (1+sqrt(5))/4*sqrt((6-2*sqrt(5)+sqrt(2*sqrt(5)-2))/(10*Pi*n)) * ((3+sqrt(5))/2+sqrt(2+2*sqrt(5)))^n
%F A219672 D-finite Recurrence: (n-1)*n*(13*n^2 - 52*n + 49)*a(n) = 3*(n-1)*(2*n-5)*(13*n^2 - 26*n + 10)*a(n-1) - (7*n^2-14*n+6)*(13*n^2 - 52*n + 49)*a(n-2) + (n-2)*(182*n^3 - 819*n^2 + 1050*n - 351)*a(n-3) - (n-3)*(n-2)*(13*n^2 - 26*n + 10)*a(n-4)
%F A219672 a(n) = (hypergeom([-n,-n], [1], phi) - hypergeom([-n,-n], [1], 1-phi))/sqrt(5) = ((1-phi)^n * P_n(-sqrt(5)-2) - phi^n * P_n(sqrt(5)-2))/sqrt(5), where phi = (1+sqrt(5))/2, P_n(x) is the Legendre polynomial. - _Vladimir Reshetnikov_, Sep 28 2016
%t A219672 Table[Sum[Binomial[n, k]^2*Fibonacci[k], {k, 0, n}], {n, 0, 20}]
%t A219672 FullSimplify@Table[((1 - GoldenRatio)^n LegendreP[n, -Sqrt[5] - 2] - GoldenRatio^n LegendreP[n, Sqrt[5] - 2])/Sqrt[5], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 28 2016 *)
%Y A219672 Cf. A000045, A001906, A219673.
%K A219672 nonn
%O A219672 0,3
%A A219672 _Vaclav Kotesovec_, Nov 24 2012