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
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
KEYWORD
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