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

0,7

Comments

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

Links

Table of n, a(n) for n=0..74.

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.

Formula

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

Example

.............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

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 *)

Crossrefs

Cf. A007318, A000108.

Sequence in context: A308400 A369816 A236541 * A158800 A144024 A185249

Adjacent sequences: A113203 A113204 A113205 * A113207 A113208 A113209

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane, Jan 07 2006

Status

approved