A278618 a(n) = Sum_{j=0..n/2} binomial(n-j-1,n-2*j)*binomial(2*n+1,j).

1, 0, 5, 7, 45, 121, 533, 1800, 7157, 26239, 101640, 384583, 1483925, 5693247, 22013059, 85076183, 330014421, 1281349195, 4985766650, 19422653367, 75775163028, 295953650376, 1157212653030, 4529183513913, 17743019073381, 69565441895001

Offset: 0

Author

Vladimir Kruchinin, Nov 23 2016

Keywords

• nonn

Links

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

Crossrefs

Cf. A005043, A055113.

Programs

Mathematica

Table[Sum[Binomial[n - j - 1, n - 2*j]*Binomial[2*n + 1, j], {j, 0, n/2}], {n,0,50}] (* G. C. Greubel, Jun 06 2017 *)

Maxima

A(x):=(12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x));
B(x):=1/((x+1)*sqrt(-3*x^2-2*x+1));
C(x):=sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4;
taylor(x*A(x)/C(x)*B(C(x)),x,0,20);

PARI

for(n=0,25, print1(sum(j=0,n, binomial(n-j-1,n-2*j)*binomial(2*n+1,j)), ", ")) \\ G. C. Greubel, Jun 06 2017

Formula

G.f.: x*A(x)/C(x)*B(C(x)), where

A(x) = (12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)),

B(x) = 1/((x+1)*sqrt(-3*x^2-2*x+1)),

C(x) = sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4.

a(n) ~ (1 - 1/sqrt(5)) * 4^n / sqrt(Pi*n). - Vaclav Kotesovec, Nov 24 2016

a(n) = (2*n + 1)*3F2(1-n/2,3/2-n/2,-2*n; 2,2-n; 4) for n>1. - Ilya Gutkovskiy, Nov 24 2016

Conjecture: 2*n*(5*n-8)*(2*n-1)*(n+1)*a(n) -n*(115*n^3-344*n^2+299*n-82)*a(n-1) -4*(2*n-1)*(5*n^3+27*n^2-74*n+30)*a(n-2) +36*(n-2)*(5*n-3)*(2*n-1)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Dec 02 2016