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A144644 Triangle in A144643 read by columns downwards.

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