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

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

Offset

0,2

Links

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.

Formula

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

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A000045, A219672, A219673.

Sequence in context: A034698 A115605 A289719 * A349769 A358963 A055836

Adjacent sequences: A114195 A114196 A114197 * A114199 A114200 A114201

Keyword

easy,nonn

Author

Paul Barry, Nov 16 2005

Status

approved