A099575 Number triangle T(n,k) = binomial(n + floor(k/2) + 1, n + 1), 0 <= k <= n.

1, 1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 6, 21, 1, 1, 7, 7, 28, 28, 1, 1, 8, 8, 36, 36, 120, 1, 1, 9, 9, 45, 45, 165, 165, 1, 1, 10, 10, 55, 55, 220, 220, 715, 1, 1, 11, 11, 66, 66, 286, 286, 1001, 1001, 1, 1, 12, 12, 78, 78, 364, 364, 1365, 1365, 4368, 1, 1, 13, 13, 91, 91, 455, 455, 1820, 1820, 6188, 6188

Offset: 0

Author

Paul Barry, Oct 23 2004

Keywords

• easy
• nonn
• tabl

Comments

Original name was: "Number triangle T(n,k) = if(k<=n, Sum_{j=0..floor(k/2)} binomial(n+j,j), 0)."

Examples

			Rows start:
  1;
  1, 1;
  1, 1,  4;
  1, 1,  5,  5;
  1, 1,  6,  6, 21;
  1, 1,  7,  7, 28, 28;
  1, 1,  8,  8, 36, 36, 120;
  1, 1,  9,  9, 45, 45, 165, 165;
  1, 1, 10, 10, 55, 55, 220, 220, 715;
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10010 (Rows 0..140, flattened)

Crossrefs

Cf. A099573, A099576 (row sums), A099577 (diagonal sums), A099578 (main diagonal).

Programs

Magma

[Binomial(n+1+Floor(k/2), n+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 24 2022

Maple

for n from 0 to 20 do seq(binomial(n+floor(k/2)+1,n+1),k=0..n) od; # Robert Israel, May 08 2018

Mathematica

Table[Binomial[n+Floor[k/2]+1, n+1], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 24 2022 *)

SageMath

flatten([[binomial(n+(k//2)+1, n+1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 24 2022

Formula

T(n, k) = binomial(n + floor(k/2) + 1, n + 1).

T(n, n) = A099578(n).

Sum_{k=0..n} T(n, k) = A099576(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A099577(n).

Extensions

Definition simplified by Robert Israel, May 08 2018