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A151509 The triangle in A151338 read by rows downwards.

%I A151509 #29 Mar 28 2020 16:30:44
%S A151509 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,
%T A151509 56,301,350,140,21,1,0,0,91,938,1701,1050,266,28,1,0,0,126,2737,7686,
%U A151509 6951,2646,462,36,1,0,0,126,7455,32725,42315,22827,5880,750,45,1,0,0,0,18711,132055
%N A151509 The triangle in A151338 read by rows downwards.
%C A151509 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
%H A151509 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.
%H A151509 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009 (see Table 6 E4(n,k) page 15).
%F A151509 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
%e A151509 Triangle begins:
%e A151509   1;
%e A151509   0, 1;
%e A151509   0, 1,  1;
%e A151509   0, 1,  3,   1;
%e A151509   0, 1,  7,   6,    1;
%e A151509   0, 1, 15,  25,   10,    1;
%e A151509   0, 0, 31,  90,   65,   15,   1;
%e A151509   0, 0, 56, 301,  350,  140,  21,  1;
%e A151509   0, 0, 91, 938, 1701, 1050, 266, 28, 1;
%t A151509 rows = 10;
%t A151509 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}]];
%t A151509 B = BellMatrix[If[# < 5, 1, 0]&, rows];
%t A151509 Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 14 2018, after _Peter Luschny_ *)
%o A151509 (Sage) # uses[bell_matrix from A264428]
%o A151509 bell_matrix(lambda n: 1 if n<5 else 0, 12) # _Peter Luschny_, Jan 19 2016
%Y A151509 Cf. A110038 (row sums), A122848, A111246, A144644, A151511.
%K A151509 nonn,tabl,easy
%O A151509 0,9
%A A151509 _N. J. A. Sloane_, May 14 2009
%E A151509 Row 9 added by _Michel Marcus_, Feb 13 2014
%E A151509 More rows from _R. J. Mathar_, May 28 2019