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

0,8

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

Links

Table of n, a(n) for n=0..65.

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).

Example

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

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 *)

Crossrefs

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

Sequence in context: A085791 A144645 A151510 * A106800 A308484 A227320

Adjacent sequences: A151509 A151510 A151511 * A151513 A151514 A151515

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, May 14 2009

Extensions

Row 9 added by Michel Marcus, Feb 13 2014

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

Status

approved