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 A137211 Generalized or s-Catalan numbers. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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 LINKS Table of n, a(n) for n=1..44. 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. FORMULA T(n,m) = binomial(m*n,n)/((m-1)*n+1). EXAMPLE {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} 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] CROSSREFS Sequence in context: A185997 A231733 A182822 * A212275 A189036 A319952 Adjacent sequences: A137208 A137209 A137210 * A137212 A137213 A137214 KEYWORD nonn AUTHOR Roger L. Bagula, Mar 05 2008 EXTENSIONS Edited by N. J. A. Sloane, May 16 2008 STATUS approved

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Last modified December 8 14:30 EST 2023. Contains 367679 sequences. (Running on oeis4.)