A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).

1, 4, 1, 24, 12, 1, 192, 144, 24, 1, 1920, 1920, 480, 40, 1, 23040, 28800, 9600, 1200, 60, 1, 322560, 483840, 201600, 33600, 2520, 84, 1, 5160960, 9031680, 4515840, 940800, 94080, 4704, 112, 1, 92897280, 185794560, 108380160, 27095040, 3386880, 225792

Offset: 1

Author

Vladeta Jovovic, Jan 29 2003

Keywords

• nonn
• tabl

Comments

Also the Bell transform of A002866(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

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, -1 <= d <= 2). - Shuhei Tsujie, Apr 26 2019

Examples

			Triangle begins:
     1;
     4,    1;
    24,   12,   1;
   192,  144,  24,  1;
  1920, 1920, 480, 40, 1;
  ...
		

Links

• P. Bala, The white diamond product of power series.
• 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. A002866 (first column), A025168 (row sums).

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 2^n*(n+1)!, 9); # Peter Luschny, Jan 26 2016

Mathematica

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2^#*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Formula

E.g.f.: exp(x*y/(1-2*x)).

T(n, k) = n!/k!*binomial(n-1, k-1)*2^(n-k). - Vladeta Jovovic, Sep 24 2003

The n-th row polynomial equals x o (x + 2) o (x + 4) o ... o (x + 2*n), where o is the deformed Hadamard product of power series defined in Bala, section 3.1. - Peter Bala, Jan 18 2018