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%I #37 Aug 28 2019 15:41:32
%S 1,7,1,91,21,1,1729,511,42,1,43225,15015,1645,70,1,1339975,523705,
%T 69300,4025,105,1,49579075,21240765,3226405,230300,8330,147,1,
%U 2131900225,984172735,166428990,13820205,621810,15386,196,1,104463111025
%N Triangle of numbers related to triangle A092083; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
%C a(n,m) := S2(7; n,m) is the seventh triangle of numbers in the sequence S2(k;n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, respectively. a(n,1)=A008542(n), n>=1.
%C a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 7-ary trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - _Wolfdieter Lang_, Sep 14 2007
%C Also the Bell transform of A008542(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%H F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://dx.doi.org/10.1007/3-540-55251-0_2">Varieties of Increasing Trees</a>, in Lecture Notes in Computer Science vol. 581, (1992), pp. 24-48.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>, arXiv:quant-phys/0402027, 2004.
%H 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 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 W. Lang, <a href="/A092082/a092082.txt">First 10 rows</a>.
%H W. Lang, <a href="http://www.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 Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012
%F a(n, m) = sum(|A051151(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. with _Wolfdieter 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.
%F a(n, m) = n!*A092083(n, m)/(m!*6^(n-m)); a(n+1, m) = (6*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 E.g.f. for m-th column: ((-1+(1-6*x)^(-1/6))^m)/m!.
%e {1}; {7,1}; {91,21,1}; {1729,511,42,1}; ...
%p # The function BellMatrix is defined in A264428.
%p # Adds (1, 0, 0, 0, ..) as column 0.
%p BellMatrix(n -> mul(6*k+1, k=0..n), 9); # _Peter Luschny_, Jan 26 2016
%t mmax = 9; a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 6*x)^(-1/6))^m)/m!, {x, 0, mmax}], x^n];
%t Flatten[Table[a[n, m], {n, 1, mmax}, {m, 1, n}]][[1 ;; 37]] (* _Jean-François Alcover_, Jun 22 2011, after e.g.f. *)
%t rows = 9;
%t t = Table[Product[6k+1, {k, 0, n}], {n, 0, rows}];
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%Y Cf. A092084 (row sums), A092085 (alternating row sums).
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Mar 19 2004
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