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

0,5

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 = 6.

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A229031 A221406 A267428 * A326798 A090217 A267570

Adjacent sequences: A156736 A156737 A156738 * A156740 A156741 A156742

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

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

Status

approved