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

Author

Paul Barry, Oct 23 2004

Keywords

• easy
• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A029907, A099575, A099577.

Programs

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

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 *)

PARI

a(n) = sum(k=0, n, sum(j=0, floor(k/2), binomial(n+j, j))); \\ Andrew Howroyd, Feb 13 2018

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

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