A219673 a(n) = Sum_{k=0..n} binomial(n,k)^2*Lucas(k) where Lucas(n) = A000032(n).

2, 3, 9, 42, 197, 913, 4302, 20611, 99773, 486438, 2385319, 11752931, 58139858, 288572079, 1436398329, 7167499522, 35842352013, 179576501169, 901226053422, 4529717794607, 22797936691207, 114881558737498, 579544350869889, 2926592507364717, 14792448049794122

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Legendre Polynomial.

Formula

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).

a(n) ~ (1+sqrt(5))/4*sqrt((6-2*sqrt(5)+sqrt(2*sqrt(5)-2))/(2*Pi*n)) * ((3+sqrt(5))/2+sqrt(2+2*sqrt(5)))^n.

Recurrence (same as for A219672): (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) = phi^n * P_n(sqrt(5)-2) + (1-phi)^n * P_n(-sqrt(5)-2), where phi = (1+sqrt(5))/2, P_n(x) is the Legendre polynomial. - Vladimir Reshetnikov, Sep 28 2016

Mathematica

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

Crossrefs

Cf. A000032, A005248, A219672.

Sequence in context: A162093 A052828 A214915 * A012972 A316916 A162094

Adjacent sequences: A219670 A219671 A219672 * A219674 A219675 A219676

Keyword

nonn

Author

Vaclav Kotesovec, Nov 24 2012

Status

approved