A113206 Triangle read by rows of generalized Catalan numbers.

1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 3, 5, 3, 0, 1, 1, 0, 4, 12, 14, 12, 4, 0, 1, 1, 0, 5, 22, 55, 42, 55, 22, 5, 0, 1, 1, 0, 6, 35, 140, 273, 132, 273, 140, 35, 6, 0, 1, 1, 0, 7, 51, 285, 969, 1428, 429, 1428, 969, 285, 51, 7, 0, 1, 1, 0, 8, 70, 506, 2530, 7084, 7752, 1430, 7752, 7084

Offset: 0

Author

N. J. A. Sloane, Jan 07 2006

Keywords

• nonn
• tabf
• easy

Comments

A dual to Pascal's triangle. Row n has 2n+1 entries.

Examples

			.............1
...........1.0.1
.........1.0.2.0.1
.......1.0.3.5.3.0.1
....1.0.4.12.14.12.4.0.1
.1.0.5.22.55.42.55.22.5.0.1
		

Links

• Ph. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers, arXiv:math/0512437 [math.CO], 2005.
• Ph. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers, J. Integer Seqs., 10 (2007), #07.4.7.

Crossrefs

Cf. A007318, A000108.

Programs

Maple

A070914 := proc(n,k) binomial(n*(k+1),n)/(n*k+1) ; end proc:
A113206 := proc(n,k) if k = 2 or k = 2*n-2 then 0 ; else A070914(n-abs(n-k)-1,abs(n-k)+1) ; fi ; end proc:
for n from 0 to 10 do for k from 1 to 2*n-1 do printf("%d ",A113206(n,k)) ; od: od: # R. J. Mathar, Feb 08 2008

Mathematica

A070914[n_, k_] := Binomial[n*(k + 1), n]/(n*k + 1);
A113206[n_, k_] := If[k == 2 || k == 2*n - 2, 0, A070914[n - Abs[n-k] - 1, Abs[n-k] + 1]];
Table[A113206[n, k], {n, 0, 10}, {k, 1, 2*n - 1}] // Flatten (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar *)

Formula

T(n,k) = A070914(n-|n-k|-1,|n-k|+1) if 3<=k<=2n-3 . - R. J. Mathar, Feb 08 2008