A151512 The triangle in A151359 read by rows upwards.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 0, 0, 1, 28, 266, 1050, 1701, 966, 119, 0, 0, 1, 36, 462, 2646, 6951, 7770, 2989, 210, 0, 0, 1, 45, 750, 5880, 22827, 42525, 33985, 8925, 336, 0, 0

Offset: 0

Author

N. J. A. Sloane, May 14 2009

Keywords

• nonn
• tabl
• easy

Comments

Conjectured: The i-th element of row j is the number of different equivalence relationships, within a set of (j-1) element, having (j-i) equivalence classes. For example: row 5 = [1, 6, 7, 1, 0] means that, in a set of 4 elements, there exists 7 equivalence relationships having 3 different equivalence classes. - Philippe Beaudoin, Nov 09 2013

Examples

			Triangle begins:
  1
  1  0
  1  1   0
  1  3   1    0
  1  6   7    1    0
  1 10  25   15    1   0
  1 15  65   90   31   1   0
  1 21 140  350  301  63   0 0
  1 28 266 1050 1701 966 119 0 0
		

Links

• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009) (see Table 7 E5(n,k) page 16).

Crossrefs

Cf. A148092 (row sums), A151511 (row-reversed).

Programs

Mathematica

Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; t[n, k_] := t[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n - 1, j - 1] a[j] t[n - j, k - 1], {j, 0, n - k + 1}]]; Table[Table[t[n - 1, k], {k, n - 1, 0, -1}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 20 2016, using Peter Luschny's Bell transform *)

Extensions

Row 9 added by Michel Marcus, Feb 13 2014

Row 10 from R. J. Mathar, May 28 2019