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A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

%I A132056 #25 Aug 28 2019 15:44:17
%S A132056 1,8,1,120,24,1,2640,672,48,1,76560,22800,2160,80,1,2756160,920160,
%T A132056 104880,5280,120,1,118514880,43243200,5639760,347760,10920,168,1,
%U A132056 5925744000,2323918080,336510720,24071040,937440,20160,224,1
%N A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.
%C A132056 Previous name was: Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
%C A132056 a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 8-ary trees. See the F. Bergeron et al. reference, especially Table 1, first row, for the e.g.f. for m=1.
%C A132056 a(n,m) := S2(8; n,m) is the eighth triangle of numbers in the sequence S2(k;n,m), k=1..7: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, A092082, respectively. a(n,1)=A045754(n), n>=1.
%H A132056 F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of Increasing Trees</a>, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
%H A132056 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.
%H A132056 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H A132056 W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A132056 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3
%H A132056 W. Lang, <a href="/A132056/a132056.txt">First 10 rows</a>.
%F A132056 a(n, m) = n!*A132057(n, m)/(m!*7^(n-m)); a(n+1, m) = (7*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1;
%F A132056 E.g.f. of m-th column: ((-1+(1-7*x)^(-1/7))^m)/m!.
%F A132056 a(n, m) = sum(|A051186(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m):= (j, m) (Stirling2 triangle). Priv. comm. with W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
%e A132056 {1}; {8,1}; {120,24,1}; {2640,672,48,1}; ...
%p A132056 # The function BellMatrix is defined in A264428.
%p A132056 # Adds (1,0,0,0, ..) as column 0.
%p A132056 BellMatrix(n -> mul(7*k+1, k=0..n), 8); # _Peter Luschny_, Jan 27 2016
%t A132056 a[n_, m_] := a[n, m] = ((m*a[n-1, m-1]*(m-1)! + (m+7*n-7)*a[n-1, m]*m!)*n!)/(n*m!*(n-1)!);
%t A132056 a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1;
%t A132056 Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]]
%t A132056 (* _Jean-François Alcover_, Jun 17 2011 *)
%t A132056 rows = 8;
%t A132056 a[n_, m_] := BellY[n, m, Table[Product[7k+1, {k, 0, j}], {j, 0, rows}]];
%t A132056 Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%Y A132056 Cf. A132060 (row sums), A132061 (alternating row sums).
%Y A132056 Cf. A092082 S2(7) triangle.
%K A132056 nonn,easy,tabl
%O A132056 1,2
%A A132056 _Wolfdieter Lang_ Sep 14 2007
%E A132056 New name from _Peter Luschny_, Jan 27 2016