A092082 Triangle of numbers related to triangle A092083; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...

1, 7, 1, 91, 21, 1, 1729, 511, 42, 1, 43225, 15015, 1645, 70, 1, 1339975, 523705, 69300, 4025, 105, 1, 49579075, 21240765, 3226405, 230300, 8330, 147, 1, 2131900225, 984172735, 166428990, 13820205, 621810, 15386, 196, 1, 104463111025

Offset

1,2

Comments

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.

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

Also the Bell transform of A008542(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Links

Table of n, a(n) for n=1..37.

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, (1992), pp. 24-48.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

W. Lang, First 10 rows.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012

Formula

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.

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.

E.g.f. for m-th column: ((-1+(1-6*x)^(-1/6))^m)/m!.

Example

{1}; {7,1}; {91,21,1}; {1729,511,42,1}; ...

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(6*k+1, k=0..n), 9); # Peter Luschny, Jan 26 2016

Mathematica

mmax = 9; a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 6*x)^(-1/6))^m)/m!, {x, 0, mmax}], x^n];
Flatten[Table[a[n, m], {n, 1, mmax}, {m, 1, n}]][[1 ;; 37]] (* Jean-François Alcover, Jun 22 2011, after e.g.f. *)
rows = 9;
t = Table[Product[6k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

Crossrefs

Cf. A092084 (row sums), A092085 (alternating row sums).

Sequence in context: A119935 A027447 A027517 * A013559 A051186 A012034

Adjacent sequences: A092079 A092080 A092081 * A092083 A092084 A092085

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 19 2004

Status

approved