A132461 Row squared sums of triangle of Lucas polynomials (A034807) for n>0: Sum_{k=0..floor(n/2)} A034807(n,k)^2, with a(0)=1.

1, 1, 5, 10, 21, 51, 122, 295, 725, 1792, 4455, 11133, 27930, 70305, 177483, 449160, 1139157, 2894625, 7367720, 18781387, 47941271, 122524216, 313484385, 802877055, 2058184346, 5280670051, 13559216117, 34841384560, 89587774395

Offset: 0

Author

Paul D. Hanna, Aug 21 2007

Keywords

• nonn

Comments

Also equals row squared sums of triangle A132460 and so equals the sum of the initial floor(n/2)+1 squared terms of 1/C(x)^n where C(x) is the g.f. of the Catalan numbers (A000108).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Marcos Mariño and Claudia Rella, On the structure of wave functions in complex Chern-Simons theory, arXiv:2312.00624 [hep-th], 2023. See p. 23.

Crossrefs

Cf. A034807 (Lucas polynomials); A093128, A132460, A132459; A000108 (Catalan).

Programs

Mathematica

Flatten[{1,Table[Sum[(Binomial[n-k,k] + Binomial[n-k-1,k-1])^2,{k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)

PARI

{a(n)=sum(k=0,n\2,(binomial(n-k, k)+binomial(n-k-1, k-1))^2)}
for(n=0, 30, print1(a(n), ", "))

PARI

/* squared sums of negative powers of Catalan series: */
{a(n)=local(Catalan=2/(1+sqrt(1-4*x +x*O(x^n)))); sum(k=0,n\2,polcoeff(Catalan^-n,k)^2)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..floor(n/2)} ( C(n-k, k) + C(n-k-1, k-1) )^2.

Ignoring initial term, equals the logarithmic derivative of A093128, which gives the number of dissections of a polygon using strictly disjoint diagonals. - Paul D. Hanna, Nov 09 2013

From Vaclav Kotesovec, Feb 28 2014: (Start)

Recurrence (for n>=5): (n-3)*n*a(n) = (2*n^2 - 7*n + 4)*a(n-1) + (n-4)*n*a(n-2) + (2*n^2 - 9*n + 8)*a(n-3) - (n-4)*(n-1)*a(n-4).

G.f.: (2-x+2*x^2)/sqrt((x^2+x+1)*(x^2-3*x+1))-1.

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

(End)