A320824 T(n, k) = (m*n)!/(k!*(n-k)!)^m with m = 3; triangle read by rows, 0 <= k <= n.

1, 6, 6, 90, 720, 90, 1680, 45360, 45360, 1680, 34650, 2217600, 7484400, 2217600, 34650, 756756, 94594500, 756756000, 756756000, 94594500, 756756, 17153136, 3705077376, 57891834000, 137225088000, 57891834000, 3705077376, 17153136

Offset

0,2

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

Formula

T(n, k) = ((3*n)!/(n!)^3) * binomial(n, k)^3 = A006480(n)*A181543(n, k).

Example

Triangle starts:
[0]      1;
[1]      6,        6;
[2]     90,      720,        90;
[3]   1680,    45360,     45360,      1680;
[4]  34650,  2217600,   7484400,   2217600,    34650;
[5] 756756, 94594500, 756756000, 756756000, 94594500, 756756;

Maple

T := (n, k, m) -> (m*n)!/(k!*(n-k)!)^m:
seq(seq(T(n, k, 3), k=0..n), n=0..7);

Mathematica

Table[((3*n)!/(n!)^3)*Binomial[n, k]^3, {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 27 2018 *)

Prog

(PARI) t(n, k) = (3*n)!/(k!*(n-k)!)^3
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 6 rows of triangle as follows: */
trianglerows(6) \\ Felix Fröhlich, Oct 21 2018
(Magma) [[(Factorial(3*n)/(Factorial(n))^3)*Binomial(n, k)^3: k in [0..n]]: n in [0..15]]; // G. C. Greubel, Oct 27 2018
(GAP) Flat(List([0..6], n->List([0..n], k->Factorial(3*n)/(Factorial(k)*Factorial(n-k))^3))); # Muniru A Asiru, Oct 27 2018

Crossrefs

Cf. A007318 (Pascal, m=1), A069466 (m=2), this sequence (m=3).

Cf. A006480, A181543.

Sequence in context: A146892 A347916 A361738 * A085804 A012125 A267139

Adjacent sequences: A320821 A320822 A320823 * A320825 A320826 A320827

Keyword

nonn,tabl

Author

Peter Luschny, Oct 21 2018

Status

approved