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%I #22 Mar 28 2020 17:07:51
%S 1,0,1,0,1,1,0,1,3,1,0,1,7,6,1,0,1,15,25,10,1,0,1,31,90,65,15,1,0,0,
%T 63,301,350,140,21,1,0,0,119,966,1701,1050,266,28,1,0,0,210,2989,7770,
%U 6951,2646,462,36,1,0,0,336,8925,33985,42525,22827,5880,750,45,1,0,0,462,25641
%N The triangle in A151359 read by rows downwards.
%C The Bell transform of g(n) = 1 if n<6 else 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%H 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, 2017.
%H 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 7 E5(n,k) page 16).
%F Bivariate e.g.f. A151511(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G6(x)), where G6(x) = Sum_{i=1..6} x^i/i! is the e.g.f. of column 1. - _R. J. Mathar_, May 28 2019
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 3 1
%e 0 1 7 6 1
%e 0 1 15 25 10 1
%e 0 1 31 90 65 15 1
%e 0 0 63 301 350 140 21 1
%e 0 0 119 966 1701 1050 266 28 1
%t 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[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 20 2016, after _Peter Luschny_ *)
%o (Sage) # uses[bell_matrix from A264428]
%o bell_matrix(lambda n: 1 if n<6 else 0, 12) # _Peter Luschny_, Jan 19 2016
%Y Begins in same way as triangle of Stirling numbers of second kind, A048993, but is strictly different. _N. J. A. Sloane_, Aug 09 2017
%Y Cf. A148092 (row sums), A122848, A111246, A144644, A151509.
%K nonn,tabl,easy
%O 0,9
%A _N. J. A. Sloane_, May 14 2009
%E Row 9 added by _Michel Marcus_, Feb 13 2014
%E More rows from _R. J. Mathar_, May 28 2019
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