A055836 T(2n+2, n), where T is the array in A055830.

2, 7, 31, 145, 701, 3458, 17298, 87417, 445225, 2281565, 11750245, 60763950, 315315014, 1641046720, 8562466432, 44775095601, 234594444741, 1231249999640, 6472043549400, 34067089542255, 179543120927115

Offset: 0

Author

Clark Kimberling, May 28 2000

Keywords

• nonn

Links

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

Programs

Magma

[Binomial(2*n+1, n) + (&+[Binomial(j+1, n-j+1)*Binomial(n+j, n): j in [Ceiling(n/2)..n]]): n in [0..25]]; // G. C. Greubel, Jun 09 2019

Mathematica

a[n_]:= Binomial[2n+1, n] + Sum[Binomial[i+1, n-i+1] Binomial[n+i, n], {i, Ceiling[n/2], n}];
Array[a, 21, 0]  (* Jean-François Alcover, Jun 03 2019, after Vladimir Kruchinin *)

Maxima

a(n):=binomial(2*n+1,n)+sum(binomial(i+1,n-i+1)*binomial(n+i,n),i,ceiling((n)/2),n); /* Vladimir Kruchinin, Nov 26 2014 */

PARI

{a(n) = binomial(2*n+1,n) + sum(j=ceil(n/2), n, binomial(j+1, n-j+1)*binomial(n+j,n))}; \\ G. C. Greubel, Jun 09 2019

Sage

def A055836(n):
    c = ceil(n/2)
    b = binomial(c+1,n-c+1)*binomial(n+c,n)
    h = hypergeometric([1,c+2,-n+c-1,n+c+1],[c+1,-n/2+c+1/2,-n/2+c+1],-1/4)
    return b*h.simplify_hypergeometric()
[A055836(n) for n in range(21)] # Peter Luschny, Nov 28 2014

Formula

a(n) = binomial(2*n+1,n) + Sum_{i=ceiling(n/2)..n} binomial(i+1,n-i+1)*binomial(n+i,n). - Vladimir Kruchinin, Nov 26 2014

a(n) = C(c+1,n-c+1)*C(n+c,n)*hypergeom([1,c+2,-n+c-1,n+c+1],[c+1,-n/2+c+1/2,-n/2+c+1],-1/4) where c=ceiling(n/2). - Peter Luschny, Nov 28 2014

Conjecture: 5*n*(n+1)*(7*n-5)*a(n) - n*(154*n^2+2*n-77)*a(n-1) - 3*(3*n-4)*(7*n+2)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Mar 13 2016