A055836 - OEIS

%I #27 Sep 08 2022 08:45:01

%S 2,7,31,145,701,3458,17298,87417,445225,2281565,11750245,60763950,

%T 315315014,1641046720,8562466432,44775095601,234594444741,

%U 1231249999640,6472043549400,34067089542255,179543120927115

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

%H G. C. Greubel, <a href="/A055836/b055836.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%F 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

%F 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

%t a[n_]:= Binomial[2n+1, n] + Sum[Binomial[i+1, n-i+1] Binomial[n+i, n], {i, Ceiling[n/2], n}];

%t Array[a, 21, 0]  (* _Jean-François Alcover_, Jun 03 2019, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o 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 */

%o (Sage)

%o def A055836(n):

%o     c = ceil(n/2)

%o     b = binomial(c+1,n-c+1)*binomial(n+c,n)

%o     h = hypergeometric([1,c+2,-n+c-1,n+c+1],[c+1,-n/2+c+1/2,-n/2+c+1],-1/4)

%o     return b*h.simplify_hypergeometric()

%o [A055836(n) for n in range(21)] # _Peter Luschny_, Nov 28 2014

%o (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

%o (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

%K nonn

%O 0,1

%A _Clark Kimberling_, May 28 2000