A098172 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 4, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 1, 20, 1, 0, 0, 0, 0, 0, 7, 35, 1, 0, 0, 0, 0, 0, 0, 28, 56, 1, 0, 0, 0, 0, 0, 0, 1, 84, 84, 1, 0, 0, 0, 0, 0, 0, 0, 10, 210, 120, 1, 0, 0, 0, 0, 0, 0, 0, 0, 55, 462, 165, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 220, 924, 220, 1

Offset: 0

Author

Paul Barry, Aug 30 2004

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A024493.

From R. J. Mathar, Mar 22 2013: (Start)

The matrix inverse starts

1;

0, 1;

0, 0, 1;

0, 0, -1, 1;

0, 0, 4, -4, 1;

0, 0, -40, 40, -10, 1;

0, 0, 796, -796, 199, -20, 1;

0, 0, -27580, 27580, -6895, 693, -35, 1;

... (End)

Examples

			Rows begin
  {1},
  {0,1},
  {0,0,1},
  {0,0,1,1},
  {0,0,0,4,1},
  {0,0,0,0,10,1},
  ...
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened

Crossrefs

Cf. A098158.

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 3*(n-k)) ))); # G. C. Greubel, Mar 15 2019

Magma

[[Binomial(n, 3*(n-k)): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Mar 15 2019

Mathematica

Table[Binomial[n, 3(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 15 2019 *)

PARI

{T(n, k) = binomial(n, 3*(n-k))}; \\ G. C. Greubel, Mar 15 2019

Sage

[[binomial(n, 3*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 15 2019

Formula

Triangle T(n, k) = binomial(n, 3(n-k)).