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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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: A157881 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

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Last modified October 2 06:08 EDT 2022. Contains 357191 sequences. (Running on oeis4.)