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%I #40 Oct 12 2023 02:01:58
%S 1,0,1,0,1,1,0,1,3,1,0,1,7,6,1,0,0,15,25,10,1,0,0,25,90,65,15,1,0,0,
%T 35,280,350,140,21,1,0,0,35,770,1645,1050,266,28,1,0,0,0,1855,6930,
%U 6825,2646,462,36,1,0,0,0,3675,26425,39795,22575,5880,750,45,1
%N Triangle in A144643 read by columns downwards.
%C The Bell transform of the sequence "g(n) = 1 if n<4 else 0". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%H G. C. Greubel, <a href="/A144644/b144644.txt">Rows n = 0..50 of the triangle, flattened</a>
%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 [math.CO], 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.
%F Bivariate e.g.f. A144644(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G4(x)), where G4(x) = Sum_{i=1..4} x^i/i! is the e.g.f. of column 1. - _R. J. Mathar_, May 28 2019
%F From _G. C. Greubel_, Oct 11 2023: (Start)
%F T(n, k) = A144643(k, n).
%F T(n, k) = A144645(n, n-k).
%F T(n, k) = t(k, n), where t(n, k) = Sum_{j=0..3} binomial(k-1, j) * t(n-1, k-j-1), with t(n,n) = 1, t(n,k) = 0 if n < 1 or n > k.
%F Sum_{k=0..n} T(n, k) = A001681(n). (End)
%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, 0, 15, 25, 10, 1;
%e 0, 0, 25, 90, 65, 15, 1;
%e 0, 0, 35, 280, 350, 140, 21, 1;
%e 0, 0, 35, 770, 1645, 1050, 266, 28, 1;
%e 0, 0, 0, 1855, 6930, 6825, 2646, 462, 36, 1;
%e 0, 0, 0, 3675, 26425, 39795, 22575, 5880, 750, 45, 1;
%e 0, 0, 0, 5775, 90475, 211750, 172095, 63525, 11880, 1155, 55, 1;
%t With[{r=15}, Table[BellY[n, k, {1,1,1,1}], {n,0,r}, {k,0,n}]]//Flatten (* _Jan Mangaldan_, May 22 2016 *)
%o (Sage) # uses[bell_matrix from A264428]
%o bell_matrix(lambda n: 1 if n<4 else 0, 12) # _Peter Luschny_, Jan 19 2016
%o (PARI) \\ Function bell_matrix is defined in A264428.
%o B = bell_matrix( n -> {if(n < 4, 1, 0)}, 9); for(n = 0, 9, printp(); for(k = 1, n, print1(B[n,k], " "))); \\ _Peter Luschny_, Apr 17 2019
%o (Magma)
%o function t(n,k)
%o if k eq n then return 1;
%o elif k le n-1 or n le 0 then return 0;
%o else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
%o end if;
%o end function;
%o A144644:= func< n,k | t(k,n) >;
%o [A144644(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 11 2023
%Y Cf. A001681 (row sums), A111246, A122848, A144643, A144645, A151509, A151511, A264428.
%K nonn,tabl
%O 0,9
%A _David Applegate_ and _N. J. A. Sloane_, Jan 25 2009
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