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A142467
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Triangle T(n,m) read by rows: T(n,m) = Product_{i=0..6} binomial(n+i,m)/binomial(m+i,m).
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16
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1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 120, 540, 120, 1, 1, 330, 4950, 4950, 330, 1, 1, 792, 32670, 108900, 32670, 792, 1, 1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1, 1, 3432, 736164, 16195608, 44537922, 16195608, 736164, 3432, 1
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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T(n,m) = A142465(n,m)*binomial(n+6,m)/binomial(m+6,m).
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 8, 1;
1, 36, 36, 1;
1, 120, 540, 120, 1;
1, 330, 4950, 4950, 330, 1;
1, 792, 32670, 108900, 32670, 792, 1;
1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1;
1, 3432, 736164, 16195608, 44537922, 16195608, 736164, 3432, 1;
1, 6435, 2760615, 131589315, 868489479, 868489479, 131589315, 2760615, 6435, 1;
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MATHEMATICA
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T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j, 0, 6}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Nov 13 2022 *)
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PROG
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(PARI) T(n, k) = prod(j=0, 6, binomial(n+j, k)/binomial(k+j, k)); \\ Seiichi Manyama, Apr 01 2021
(Magma) [(&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..6]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2022
(SageMath)
def A142467(n, k): return product(binomial(n+j, k)/binomial(k+j, k) for j in (0..6))
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CROSSREFS
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Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited by the Associate Editors of the OEIS, May 17 2009
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STATUS
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approved
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