A090683 Triangle read by rows, defined by T(n,k) = C(n,k)*S2(n,k), 0 <= k <= n, where C(n,k) are the binomial coefficients and S2(n,k) are the Stirling numbers of the second kind.

1, 0, 1, 0, 2, 1, 0, 3, 9, 1, 0, 4, 42, 24, 1, 0, 5, 150, 250, 50, 1, 0, 6, 465, 1800, 975, 90, 1, 0, 7, 1323, 10535, 12250, 2940, 147, 1, 0, 8, 3556, 54096, 119070, 58800, 7448, 224, 1, 0, 9, 9180, 254100, 979020, 875826, 222264, 16632, 324, 1, 0, 10, 22995, 1119600, 7162050, 10716300, 4793670, 705600, 33750, 450, 1

Offset: 0

Author

Philippe Deléham, Dec 18 2003

Keywords

• easy
• nonn
• tabl

Comments

T(n,k) is the number of Green's H-classes contained in the D-class of rank k in the full transformation semigroup on [n]. - Geoffrey Critzer, Dec 27 2022

Examples

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  9,  1;
  0,  4, 42, 24,  1;
  ...
		

References

• O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, pages 58-62.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..230
• Wikipedia, Green's relations
• Wikipedia, Transformation semigroup

Crossrefs

Cf. A007318, A048993, A090657.

Row sum sequence is A122455.

Programs

Mathematica

Flatten[Table[Table[Binomial[n, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 10}], 1]

Maxima

create_list(binomial(n,k)*stirling2(n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

Formula

T(n, k) = binomial(n,k)*Stirling2(n,k).

T(n, k) = A007318(n, k)*A048993(n, k).

T(n, k) = A090657(n, k)/k!.

Extensions

Edited by Olivier Gérard, Oct 23 2012