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

0,2

Comments

Diagonal sums of number triangle A115255.

Links

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

Formula

From Vaclav Kotesovec, Mar 02 2014: (Start)

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

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

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

Mathematica

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

Prog

(PARI) 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

Crossrefs

Sequence in context: A288539 A281338 A036367 * A132963 A122404 A150670

Adjacent sequences: A115253 A115254 A115255 * A115257 A115258 A115259

Keyword

easy,nonn

Author

Paul Barry, Jan 18 2006

Status

approved