A193636 Triangle: T(n,k) = C(3n-2k,k), 0 <= k <= n.

1, 1, 1, 1, 4, 1, 1, 7, 10, 1, 1, 10, 28, 20, 1, 1, 13, 55, 84, 35, 1, 1, 16, 91, 220, 210, 56, 1, 1, 19, 136, 455, 715, 462, 84, 1, 1, 22, 190, 816, 1820, 2002, 924, 120, 1, 1, 25, 253, 1330, 3876, 6188, 5005, 1716, 165, 1, 1, 28, 325, 2024, 7315, 15504, 18564

Offset: 0

Author

Clark Kimberling, Aug 01 2011

Keywords

• nonn
• tabl

Examples

			First 5 rows:
  1;
  1,  1;
  1,  4,  1;
  1,  7, 10,  1;
  1, 10, 28, 20,  1; [Corrected by _Robert Israel_, Nov 06 2018]
		

Links

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

Crossrefs

Cf. A193635.

Programs

Magma

/* As triangle */[[Binomial(3*n-2*k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 07 2018

Maple

seq(seq(binomial(3*n-2*k,k),k=0..n),n=0..10); # Robert Israel, Nov 06 2018

Mathematica

p[n_, k_] := Binomial[3 n - 2 k, k];
Table[p[n, k], {n, 0, 9}, {k, 0, n}]  (* A193636 *)
Flatten[%]

Formula

T(n,k) = C(3n-2k,k), 0 <= k <= n.

G.f. as triangle: (1-x*y)^2/(1 - x - 3*x*y + 3*x^2*y^2 - x^3*y^3). - Robert Israel, Nov 06 2018

T(n,k) = A102547(3*n,k). - R. J. Mathar, Apr 26 2024