A342972 Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).
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
Offset: 0
Examples
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;
Links
- Seiichi Manyama, Rows n = 0..50, flattened
Crossrefs
Programs
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Mathematica
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 *) -
PARI
T(n, k) = prod(j=0, n-1, binomial(n+j, k)/binomial(k+j, k));
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PARI
T(n, k) = prod(j=0, k-1, binomial(2*n-1, n+j)/binomial(2*n-1, j));
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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);
Formula
T(n,k) = Product_{j=0..k-1} binomial(2*n-1,n+j)/binomial(2*n-1,j).
Comments