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A137211
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Generalized or s-Catalan numbers.
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0
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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; internal format)
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OFFSET
| 1,5
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COMMENTS
| Comment from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 04 2008: This is a triangular section of Stanica's array of s-Catalan numbers, with rows A000108, A001764, A002293-A002296, A007556, A062994, A062744,... 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, ...
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REFERENCES
| Niederhausen, Catalan Traffic at the Beach, 2002, http://citeseer.ist.psu.edu/niederhausen02catalan.html
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LINKS
| P. Stanica, p^q-Catalan numbers and squarefree binomial coefficients, J. Numb. Theory 100 (2003) 203-216.
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FORMULA
| t(n,m)=Binomial[m*n,n]/((m-1)*n+1)
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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}
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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]
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CROSSREFS
| Sequence in context: A085853 A185997 A182822 * A189036 A174665 A083855
Adjacent sequences: A137208 A137209 A137210 * A137212 A137213 A137214
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 05 2008
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), May 16 2008
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