A219673 -id:A219673 - cp's OEIS frontend

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

0, 1, 5, 20, 87, 405, 1924, 9225, 44625, 217528, 1066725, 5256087, 26001000, 129053365, 642376709, 3205403100, 16029187391, 80309053285, 403040543420, 2025751379997, 10195547237235, 51376594943136, 259180112907875, 1308811957775785, 6615383878581072

Offset: 0

Vaclav Kotesovec, Nov 24 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200.
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Cf. A000045, A001906, A219673.

Table[Sum[Binomial[n, k]^2*Fibonacci[k], {k, 0, n}], {n, 0, 20}]
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 *)

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)
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
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)
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

A114198 a(n) = Sum_{k=0..n} binomial(n,k)^2*F(k+1).

Original entry on oeis.org

1, 2, 7, 31, 142, 659, 3113, 14918, 72199, 351983, 1726022, 8504509, 42070429, 208812722, 1039387519, 5186451311, 25935769702, 129942777227, 652133298421, 3277734587302, 16496741964221, 83129076840317, 419362231888882

Offset: 0

Paul Barry, Nov 16 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Cf. A000045, A219672, A219673.

a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 31][n+1],
     ((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-2)*(n-3)*(13*n^2-26*n+10) *a(n-4))/
      (n*(n-1)*(13*n^2-52*n+49)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 28 2016

FullSimplify@Table[(GoldenRatio^(n - 1) LegendreP[n, Sqrt[5] - 2] - (1 - GoldenRatio)^(n - 1) LegendreP[n, -Sqrt[5] - 2])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 28 2016 *)

a(n) = Sum_{k=0..n} C(n, k)^2 * F(k+1); a(n) = A114197(2n, n).
a(n) = (phi^(n-1) * P_n(sqrt(5)-2) - (1-phi)^(n-1) * P_n(-sqrt(5)-2))/sqrt(5), where phi = (1+sqrt(5))/2, P_n(x) is the Legendre polynomial.
a(n) ~ sqrt((6 + 2*sqrt(5) + sqrt(2*(29 + 13*sqrt(5))))/10)/2 * ((3 + sqrt(5))/2 + sqrt(2*(1+sqrt(5))))^n / sqrt(Pi*n). - Vaclav Kotesovec, May 06 2017
a(n) ~ sqrt(2*phi^2 + phi^(7/2)) * (2*phi^(1/2) + phi^2)^n / (2*sqrt(5*Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 22 2017
D-finite with recurrence +n*(n-1)*a(n) -5*n*(n-1)*a(n-1) +2*(-n^2+17*n-27)*a(n-2) +(11*n^2-135*n+270)*a(n-3) +2*(-17*n^2+121*n-215)*a(n-4) +(n-4)*(43*n-191)*a(n-5) -3*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, May 11 2022

cp's OEIS Frontend

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

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