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A062993
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A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.
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17
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1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 12, 4, 1, 1, 42, 55, 22, 5, 1, 1, 132, 273, 140, 35, 6, 1, 1, 429, 1428, 969, 285, 51, 7, 1, 1, 1430, 7752, 7084, 2530, 506, 70, 8, 1, 1, 4862, 43263, 53820, 23751, 5481, 819, 92, 9
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OFFSET
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0,4
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COMMENTS
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The column sequences (without leading zeros) appear in eq.(7.66), p. 347 of the Graham et al. reference, in Th. 0.3, p. 66, of Hilton and Pedersen reference, as first columns of the S-triangles in the Hoggatt and Bicknell reference and in eq. 5 of the Frey and Sellers reference. They are also called m-Raney (here m=k+2) or Fuss-Catalan sequences (see Graham et al. for reference). For the history and the name Pfaff-Fuss see Brown reference, p. 975. PF(n,m) := binomial(m*n+1,n)/(m*n+1), m >= 2.
Also called generalized Catalan numbers.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
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LINKS
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FORMULA
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a(n, k)= binomial((k+2)*(n-k), n-k)/((k+1)*(n-k)+1) = PF(n-k, k+2) if n-k >= 0, otherwise 0.
G.f. for column k: A(k, x) := x^k*RootOf(_Z^(k+2)*x-_Z+1) (Maple notation, from ECS, see links for column sequences and Graham et al. reference eq.(5.59) p. 200 and p. 349).
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EXAMPLE
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The triangle a(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 1 1
2: 2 1 1
3: 5 3 1 1
4: 14 12 4 1 1
5: 42 55 22 5 1 1
6: 132 273 140 35 6 1 1
7: 429 1428 969 285 51 7 1 1
8: 1430 7752 7084 2530 506 70 8 1 1
9: 4862 43263 53820 23751 5481 819 92 9 1 1
10: 16796 246675 420732 231880 62832 10472 1240 117 10 1 1
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MATHEMATICA
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a[n_, k_] = Binomial[(k+2)*(n-k), n-k]/((k+1)*(n-k) + 1);
Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 53]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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