A115256 Diagonal sums of correlation triangle of central binomial coefficients.

1, 2, 8, 25, 90, 312, 1145, 4186, 15640, 58681, 222298, 845848, 3235385, 12418650, 47827992, 184688185, 714884186, 2772776984, 10774163001, 41932100698, 163430680600, 637793652281, 2491918144602, 9746480252952, 38157725306425

Offset: 0

Author

Paul Barry, Jan 18 2006

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A115255.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Mathematica

CoefficientList[Series[1/((Sqrt[1-4x])(Sqrt[1-4x^2])(1-x^3)), {x,0,30}], x] (* Harvey P. Dale, Feb 15 2012 *)

PARI

my(x='x+O('x^50)); Vec(1/(sqrt(1-4*x)*sqrt(1-4*x^2)*(1-x^3))) \\ G. C. Greubel, Mar 18 2017

Formula

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x^2)*(1-x^3)).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} [j<=k]*C(2*k-2*j, k-j)*[j<=n-2*k]*C(2*n-4*k-2*j, n-2*k-j).

a(n) ~ sqrt(3) * 2^(2*n+7) / (189 * sqrt(Pi*n)). - Vaclav Kotesovec, Mar 02 2014

Conjecture: n*a(n) + 2*(-2*n+1)*a(n-1) + 4*(-n+1)*a(n-2) + 3*(5*n-8)*a(n-3) + 2*(2*n-1)*a(n-4) + 4*(n-1)*a(n-5) + 8*(-2*n+3)*a(n-6) = 0. - R. J. Mathar, Jun 22 2016