A219673 - OEIS

%I #16 May 06 2017 17:02:21

%S 2,3,9,42,197,913,4302,20611,99773,486438,2385319,11752931,58139858,

%T 288572079,1436398329,7167499522,35842352013,179576501169,

%U 901226053422,4529717794607,22797936691207,114881558737498,579544350869889,2926592507364717,14792448049794122

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

%H Vincenzo Librandi, <a href="/A219673/b219673.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>.

%F 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).

%F 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.

%F 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).

%F 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

%t Table[Sum[Binomial[n, k]^2*LucasL[k], {k, 0, n}], {n, 0, 20}]

%t 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 *)

%Y Cf. A000032, A005248, A219672.

%K nonn

%O 0,1

%A _Vaclav Kotesovec_, Nov 24 2012