A308281 The third power of the unsigned Lah triangular matrix A105278.

1, 6, 1, 54, 18, 1, 648, 324, 36, 1, 9720, 6480, 1080, 60, 1, 174960, 145800, 32400, 2700, 90, 1, 3674160, 3674160, 1020600, 113400, 5670, 126, 1, 88179840, 102876480, 34292160, 4762800, 317520, 10584, 168, 1, 2380855680, 3174474240, 1234517760, 205752960, 17146080, 762048, 18144, 216, 1

Offset: 1

Author

Shuhei Tsujie, May 18 2019

Keywords

• nonn
• tabl
• easy

Comments

Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -2 <= d <= 3).

Examples

			Triangle begins:
     1;
     6,    1;
    54,   18,    1;
   648,  324,   36,  1;
  9720, 6480, 1080, 60, 1;
  ...
		

Links

• N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Crossrefs

Cf. A105278.

Programs

Mathematica

Table[3^(n - k) * Binomial[n - 1, k - 1] * n! / k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 13 2019 *)

Formula

E.g.f.: exp(x*y/(1-3*x)).

T(n,k) = 3^(n-k)*binomial(n-1, k-1)*n!/k! = 3^(n-k)*A105278.