A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.

1, 1, 1, 1, 6, 1, 24, 1, 60, 1, 120, 360, 1, 210, 2520, 1, 336, 10080, 1, 504, 30240, 60480, 1, 720, 75600, 604800, 1, 990, 166320, 3326400, 1, 1320, 332640, 13305600, 19958400, 1, 1716, 617760, 43243200, 259459200, 1, 2184, 1081080, 121080960, 1816214400

Offset: 0

Author

Paul D. Hanna, May 07 2006

Keywords

• nonn
• tabf

Comments

Row sums form A118395.

Eigenvector is A118396.

Examples

			Triangle begins:
  1;
  1;
  1;
  1,    6;
  1,   24;
  1,   60;
  1,  120,    360;
  1,  210,   2520;
  1,  336,  10080;
  1,  504,  30240,    60480;
  1,  720,  75600,   604800;
  1,  990, 166320,  3326400;
  1, 1320, 332640, 13305600, 19958400;
  ...
		

Links

• G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Crossrefs

Cf. A118395 (row sums), A118396 (eigenvector).

Variants: A059344, A118931.

Programs

Magma

F:= Factorial;
[F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

Mathematica

T[n_, k_] := n!/(k!(n-3k)!);
Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)

PARI

T(n,k)=if(n<3*k || k<0,0,n!/k!/(n-3*k)!)

Sage

f=factorial;
flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

Formula

E.g.f.: A(x,y) = exp(x + y*x^3).