OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n+j, j).
Conjecture: 4*n*(n-1)*(3*n+2)*(n+2)*a(n) - 36*(n-1)*(n+1)*a(n-1) - 3*n*(3*n+5)*(3*n-1)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Nov 28 2014
From Robert Israel, May 08 2018: (Start)
a(2*n) = (1+n/(n+1))*binomial(3*n+1,n).
a(2*n+1) = 2*binomial(3*n+3,n).
The conjecture follows from this. (End)
a(n) = (1/(n+1))*Sum_{k=0..n} binomial(n + floor(k/2) + 1, floor(k/2) + 1)*(1 + floor(k/2)). - G. C. Greubel, Jul 24 2022
a(n) = binomial(2*n+2, n)*hypergeom([-n, n+1], [-2*n-2], -1). - Detlef Meya, Dec 25 2023
MAPLE
seq(op([(1+n/(n+1))*binomial(3*n+1, n), 2*binomial(3*n+3, n)]), n=0..20);
MATHEMATICA
a[n_] := Sum[Binomial[n + j, j], {k, 0, n}, {j, 0, k/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 06 2018 *)
a[n_] := Binomial[2*n+2, n]*Hypergeometric2F1[-n, n+1, -2*n-2, -1]; Flatten[Table[a[n], {n, 0, 28}]] (* Detlef Meya, Dec 25 2023 *)
PROG
(PARI) a(n) = sum(k=0, n, sum(j=0, floor(k/2), binomial(n+j, j))); \\ Andrew Howroyd, Feb 13 2018
(Magma) [(&+[Binomial(n+Floor(k/2)+1, Floor(k/2)+1)*(1+Floor(k/2))/(n+1): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Jul 24 2022
(SageMath) [sum( binomial(n+(k//2)+1, (k//2)+1)*(1+(k//2))/(n+1) for k in (0..n) ) for n in (0..40)] # G. C. Greubel, Jul 24 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 23 2004
STATUS
approved