A156740 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 7, read by rows.
1, 1, 1, 1, 153, 1, 1, 4845, 4845, 1, 1, 74613, 2362745, 74613, 1, 1, 735471, 358664691, 358664691, 735471, 1, 1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1, 1, 30421755, 1056158828725, 160324910200455, 160324910200455, 1056158828725, 30421755, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 153, 1; 1, 4845, 4845, 1; 1, 74613, 2362745, 74613, 1; 1, 735471, 358664691, 358664691, 735471, 1; 1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1;
Links
- G. C. Greubel, Rows n = 0..30 of the triangle, flattened
Crossrefs
Programs
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Magma
A156740:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..7]]) ) >; [A156740(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021
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Mathematica
b[n_, k_]:= Binomial[2*n, 2*k]; T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]]; Table[T[n, k, 7], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *) -
Sage
def A156740(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..7)) ) flatten([[A156740(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021
Formula
T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 7.
Sum_{k=0..n} T(n, k, 7) = A151614(n).
Extensions
Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021