A156741 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 = 8, read by rows.

1, 1, 1, 1, 190, 1, 1, 7315, 7315, 1, 1, 134596, 5181946, 134596, 1, 1, 1562275, 1106715610, 1106715610, 1562275, 1, 1, 13123110, 107904771975, 1985447804340, 107904771975, 13123110, 1, 1, 86493225, 5974000557525, 1275875833357125, 1275875833357125, 5974000557525, 86493225, 1

Offset: 0

Author

Roger L. Bagula, Feb 14 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,        1;
  1,      190,            1;
  1,     7315,         7315,             1;
  1,   134596,      5181946,        134596,            1;
  1,  1562275,   1106715610,    1106715610,      1562275,        1;
  1, 13123110, 107904771975, 1985447804340, 107904771975, 13123110, 1;
		

Links

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

Crossrefs

Cf. A086645 (m=0), A156739 (m=6), A156740 (m=7), this sequence (m=8), A156742 (m=9).

Cf. A151709 (row sums).

Programs

Magma

A156741:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..8]]) ) >;
[A156741(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 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, 8], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)

Sage

def A156741(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..8)) )
flatten([[A156741(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 = 8.

Sum_{k=0..n} T(n, k) = A151709(n).

Extensions

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021