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

0,1

Links

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

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

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 *)

Prog

(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 */
(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
(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
(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

Crossrefs

Sequence in context: A114198 A349769 A358963 * A076177 A335868 A126033

Adjacent sequences: A055833 A055834 A055835 * A055837 A055838 A055839

Keyword

nonn

Author

Clark Kimberling, May 28 2000

Status

approved