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A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

%I A088729 #13 May 05 2019 03:23:30
%S A088729 1,3,1,13,9,1,75,79,18,1,541,765,265,30,1,4683,8311,3870,665,45,1,
%T A088729 47293,100989,59101,13650,1400,63,1,545835,1362439,960498,278901,
%U A088729 38430,2618,84,1,7087261,20246445,16700545,5844510,1012431,92610,4494,108,1
%N A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.
%C A088729 Also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%C A088729 Also the number of k-dimensional flats of the n-dimensional Catalan arrangement. - _Shuhei Tsujie_, May 05 2019
%H A088729 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 A088729 E.g.f.: exp((exp(x)-1)*y/(2-exp(x))).
%p A088729 # The function BellMatrix is defined in A264428.
%p A088729 # Adds (1, 0, 0, 0, ..) as column 0.
%p A088729 BellMatrix(n -> add(combinat:-eulerian1(n+1, k)*2^k, k=0..n+1), 9); # _Peter Luschny_, Jan 26 2016
%t A088729 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A088729 rows = 12;
%t A088729 B = BellMatrix[Function[n, HurwitzLerchPhi[1/2, -n-1, 0]/2], rows];
%t A088729 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 27 2018, after _Peter Luschny_ *)
%Y A088729 Cf. A000670(first column), A075729(row sums).
%K A088729 nonn,tabl
%O A088729 1,2
%A A088729 _Vladeta Jovovic_, Nov 22 2003