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

0,5

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

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

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!.

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A007318, A048993, A090657.

Row sum sequence is A122455.

Sequence in context: A357103 A278882 A153007 * A356263 A320825 A161552

Adjacent sequences: A090680 A090681 A090682 * A090684 A090685 A090686

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Dec 18 2003

Extensions

Edited by Olivier Gérard, Oct 23 2012

Status

approved