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

0,9

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

Links

Table of n, a(n) for n=0..70.

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).

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

Example

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;

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 *)

Prog

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

Crossrefs

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

Sequence in context: A272008 A054024 A144644 * A264434 A151511 A048993

Adjacent sequences: A151506 A151507 A151508 * A151510 A151511 A151512

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, May 14 2009

Extensions

Row 9 added by Michel Marcus, Feb 13 2014

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

Status

approved