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A069270
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Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).
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4
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1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260, 1, 10
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OFFSET
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0,5
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COMMENTS
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For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).
Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019
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LINKS
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FORMULA
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T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).
For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).
T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).
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EXAMPLE
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Rows start
1;
1, 1;
1, 2, 4;
1, 3, 9, 22;
1, 4, 15, 52, 140;
etc.
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MAPLE
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binomial(n+3*k, k)*(n-k+1)/(n+2*k+1) ;
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MATHEMATICA
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Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 27 2019 *)
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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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