A099576 Row sums of triangle A099575.

1, 2, 6, 12, 35, 72, 210, 440, 1287, 2730, 8008, 17136, 50388, 108528, 319770, 692208, 2042975, 4440150, 13123110, 28614300, 84672315, 185122080, 548354040, 1201610592, 3562467300, 7821594872, 23206929840, 51037462560, 151532656696

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

Cf. A029907, A099575, A099577.

Sequence in context: A221989 A349002 A204512 * A303479 A347452 A307015

Adjacent sequences: A099573 A099574 A099575 * A099577 A099578 A099579

Keyword

easy,nonn

Author

Paul Barry, Oct 23 2004

Status

approved