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

0,5

Links

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

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).

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A151709 (row sums).

Sequence in context: A323320 A352759 A259166 * A138730 A159821 A118878

Adjacent sequences: A156738 A156739 A156740 * A156742 A156743 A156744

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

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

Status

approved