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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;
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listen;
history;
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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T(n,m) = binomial(m*n,n)/((m-1)*n+1).
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EXAMPLE
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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}
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MATHEMATICA
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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
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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