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A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).

%I A079621 #37 May 05 2019 03:23:09
%S A079621 1,4,1,24,12,1,192,144,24,1,1920,1920,480,40,1,23040,28800,9600,1200,
%T A079621 60,1,322560,483840,201600,33600,2520,84,1,5160960,9031680,4515840,
%U A079621 940800,94080,4704,112,1,92897280,185794560,108380160,27095040,3386880,225792
%N A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).
%C A079621 Also the Bell transform of A002866(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%C A079621 Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -1 <= d <= 2). - _Shuhei Tsujie_, Apr 26 2019
%H A079621 P. Bala, <a href="/A048993/a048993.pdf">The white diamond product of power series</a>.
%H A079621 N. Nakashima and S. Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%F A079621 E.g.f.: exp(x*y/(1-2*x)).
%F A079621 T(n, k) = n!/k!*binomial(n-1, k-1)*2^(n-k). - _Vladeta Jovovic_, Sep 24 2003
%F A079621 The n-th row polynomial equals x o (x + 2) o (x + 4) o ... o (x + 2*n), where o is the deformed Hadamard product of power series defined in Bala, section 3.1. - _Peter Bala_, Jan 18 2018
%e A079621 Triangle begins:
%e A079621      1;
%e A079621      4,    1;
%e A079621     24,   12,   1;
%e A079621    192,  144,  24,  1;
%e A079621   1920, 1920, 480, 40, 1;
%e A079621   ...
%p A079621 # The function BellMatrix is defined in A264428.
%p A079621 # Adds (1, 0, 0, 0, ..) as column 0.
%p A079621 BellMatrix(n -> 2^n*(n+1)!, 9); # _Peter Luschny_, Jan 26 2016
%t A079621 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A079621 B = BellMatrix[2^#*(#+1)!&, rows = 12];
%t A079621 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y A079621 Cf. A002866 (first column), A025168 (row sums).
%K A079621 nonn,tabl
%O A079621 1,2
%A A079621 _Vladeta Jovovic_, Jan 29 2003