A156739 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 = 6, read by rows.

1, 1, 1, 1, 120, 1, 1, 3060, 3060, 1, 1, 38760, 988380, 38760, 1, 1, 319770, 103285710, 103285710, 319770, 1, 1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1, 1, 9657700, 157843517260, 16494647553670, 16494647553670, 157843517260, 9657700, 1

Offset: 0

Author

Roger L. Bagula, Feb 14 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,       1;
  1,     120,          1;
  1,    3060,       3060,           1;
  1,   38760,     988380,       38760,          1;
  1,  319770,  103285710,   103285710,     319770,       1;
  1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1;
		

Links

• G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Crossrefs

Cf. A086645 (m=0), this sequence (m=6), A156740 (m=7), A156741 (m=8), A156742 (m=9).

Programs

Magma

A156739:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..6]]) ) >;
[A156739(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 18 2021

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, 6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2021 *)

Sage

def A156739(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..6)) )
flatten([[A156739(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 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 = 6.

Extensions

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 18 2021