A137211 Generalized or s-Catalan numbers.

1, 1, 1, 1, 2, 3, 1, 5, 12, 22, 1, 14, 55, 140, 285, 1, 42, 273, 969, 2530, 5481, 1, 132, 1428, 7084, 23751, 62832, 141778, 1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348, 1, 1430, 43263, 420732, 2330445, 9203634, 28989675, 77652024

Offset: 1

Author

Roger L. Bagula, Mar 05 2008

Keywords

• nonn

Comments

From R. J. Mathar, May 04 2008: (Start)

This is a triangular section of Stanica's array of s-Catalan numbers, with rows A000108, A001764, A002293-A002296, A007556, A062994, A059968,... read along diagonals in A062993 and A070914:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...

1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, ...

1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, ...

1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, ...

1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, ...

1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, ...

1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, ...

1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, ...

1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, ...

(End)

The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link for this interpretation and others), so the (k+1)-th column of Stanica's array enumerates the number of (n+1)-gon partitions of a (k*(n-1)+2)-gon. Cf. A000326 (k=3), A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014

Examples

			{1},
{1, 1},
{1, 2, 3},
{1, 5, 12, 22},
{1, 14, 55, 140, 285},
{1, 42, 273, 969, 2530, 5481},
{1, 132, 1428, 7084, 23751, 62832, 141778},
{1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348}
		

Links

• Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.
• A. Regev, The Central Component of a Triangulation, J. Int. Seq. 16 (2013) #13.4.1
• Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
• P. Stanica, p^q-Catalan numbers and squarefree binomial coefficients, J. Numb. Theory 100 (2003) 203-216.

Programs

Mathematica

t[n_, m_] := Binomial[m*n, n]/((m - 1)*n + 1); a = Table[Table[t[n, m], {m, 1, n + 1}], {n, 0, 10}]; Flatten[a]

Formula

T(n,m) = binomial(m*n,n)/((m-1)*n+1).

Extensions

Edited by N. J. A. Sloane, May 16 2008