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A342972
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Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).
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3
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1, 1, 1, 1, 3, 1, 1, 10, 10, 1, 1, 35, 105, 35, 1, 1, 126, 1176, 1176, 126, 1, 1, 462, 13860, 41580, 13860, 462, 1, 1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1, 1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1
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OFFSET
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0,5
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COMMENTS
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Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_n where (n,k)_m is Product_{j=1..k} binomial(n-j+m,m)/binomial(j-1+m,m).
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LINKS
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FORMULA
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T(n,k) = Product_{j=0..k-1} binomial(2*n-1,n+j)/binomial(2*n-1,j).
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 3, 1;
1, 10, 10, 1;
1, 35, 105, 35, 1;
1, 126, 1176, 1176, 126, 1;
1, 462, 13860, 41580, 13860, 462, 1;
1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1;
1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1;
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MATHEMATICA
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T[n_, k_] := Product[Binomial[n + i, k]/Binomial[k + i, k], {i, 0, n - 1}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 01 2021 *)
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PROG
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(PARI) T(n, k) = prod(j=0, n-1, binomial(n+j, k)/binomial(k+j, k));
(PARI) T(n, k) = prod(j=0, k-1, binomial(2*n-1, n+j)/binomial(2*n-1, j));
(PARI) f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
T(n, k) = f(n, k, n);
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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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STATUS
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approved
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