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

0,5

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 = 7.

Sum_{k=0..n} T(n, k, 7) = A151614(n).

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A086645 (m=0), A156739 (m=6), this sequence (m=7), A156741 (m=8), A156742 (m=9).

Cf. A151614 (row sums).

Sequence in context: A370312 A099117 A109778 * A095226 A346630 A165340

Adjacent sequences: A156737 A156738 A156739 * A156741 A156742 A156743

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