A156742 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 = 9, read by rows.

1, 1, 1, 1, 231, 1, 1, 10626, 10626, 1, 1, 230230, 10590580, 230230, 1, 1, 3108105, 3097744650, 3097744650, 3108105, 1, 1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1, 1, 225792840, 29367745734600, 8590065627370500, 8590065627370500, 29367745734600, 225792840, 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 = 9.

Example

Triangle begins as:
  1;
  1,        1;
  1,      231,            1;
  1,    10626,        10626,             1;
  1,   230230,     10590580,        230230,            1;
  1,  3108105,   3097744650,    3097744650,      3108105,        1;
  1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1;

Mathematica

T[n_, k_, m_]:= Round[Product[Binomial[2*(n+j), 2*(k+j)]/Binomial[2*(n-k+j), 2*j], {j, 0, m}]];
Table[T[n, k, 9], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)

Prog

(Magma)
A156742:= func< n, k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..9]]) ) >;
[A156742(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021
(Sage)
def A156742(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..9)) )
flatten([[A156742(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), A156741 (m=8), this sequence (m=9).

Sequence in context: A051183 A251275 A323321 * A031965 A316095 A345795

Adjacent sequences: A156739 A156740 A156741 * A156743 A156744 A156745

Keyword

nonn,tabl,changed

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