|
%I #40 Aug 28 2019 15:38:12
%S 1,6,1,66,18,1,1056,372,36,1,22176,9240,1200,60,1,576576,271656,42840,
%T 2940,90,1,17873856,9269568,1685376,142800,6090,126,1,643458816,
%U 360847872,73313856,7254576,386400,11256,168,1,26381811456,15799069440
%N Triangle of numbers related to triangle A049375; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297...
%C a(n,m) := S2(6; n,m) is the sixth triangle of numbers in the sequence S2(k; n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, respectively. a(n,1)= A008548(n).
%C a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 6-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
%H 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. Added Mar 01 2014.
%H F. Bergeron, Philippe Flajolet and Bruno Salvy, <a href="http://hal.inria.fr/inria-00074977">Varieties of increasing trees</a>, HAL, Rapport De Recherche Inria. Added Mar 01 2014.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%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="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 W. Lang, <a href="/A049385/a049385.txt">First 10 rows</a>.
%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]. - From _N. J. A. Sloane_, Aug 21 2012
%H E. Neuwirth, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00373-3">Recursively defined combinatorial Functions: Extending Galton's board</a>, Discr. Maths. 239 (2001) 33-51.
%F a(n, m) = n!*A049375(n, m)/(m!*5^(n-m)); a(n+1, m) = (5*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; E.g.f. of m-th column: ((-1+(1-5*x)^(-1/5))^m)/m!.
%F a(n, m) = sum(|A051150(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to _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.
%e Triangle begins:
%e {1};
%e {6,1};
%e {66,18,1};
%e {1056,372,36,1};
%e ...
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> mul(5*k+1, k=0..n), 9); # _Peter Luschny_, Jan 28 2016
%t a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 5*x)^(-1/5))^m)/m!, {x, 0, n}], x^n];
%t Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]]
%t (* _Jean-François Alcover_, Jun 21 2011, after e.g.f. *)
%t rows = 9;
%t t = Table[Product[5k+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. A049412.
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
|
