A151509 The triangle in A151338 read by rows downwards.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 0, 31, 90, 65, 15, 1, 0, 0, 56, 301, 350, 140, 21, 1, 0, 0, 91, 938, 1701, 1050, 266, 28, 1, 0, 0, 126, 2737, 7686, 6951, 2646, 462, 36, 1, 0, 0, 126, 7455, 32725, 42315, 22827, 5880, 750, 45, 1, 0, 0, 0, 18711, 132055

Offset: 0

Author

N. J. A. Sloane, May 14 2009

Keywords

• nonn
• tabl
• easy

Comments

The Bell transform of the sequence "g(n) = 1 if n < 5, otherwise 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  7,   6,    1;
  0, 1, 15,  25,   10,    1;
  0, 0, 31,  90,   65,   15,   1;
  0, 0, 56, 301,  350,  140,  21,  1;
  0, 0, 91, 938, 1701, 1050, 266, 28, 1;
		

Links

• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009 (see Table 6 E4(n,k) page 15).

Crossrefs

Cf. A110038 (row sums), A122848, A111246, A144644, A151511.

Programs

Mathematica

rows = 10;
BellMatrix[f_Function | f_Symbol, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[If[# < 5, 1, 0]&, rows];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

Sage

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<5 else 0, 12) # Peter Luschny, Jan 19 2016

Formula

Bivariate e.g.f A151509(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G5(x)), where G5(x) = Sum_{i=1..5} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

Extensions

Row 9 added by Michel Marcus, Feb 13 2014

More rows from R. J. Mathar, May 28 2019