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

1, 8, 1, 120, 24, 1, 2640, 672, 48, 1, 76560, 22800, 2160, 80, 1, 2756160, 920160, 104880, 5280, 120, 1, 118514880, 43243200, 5639760, 347760, 10920, 168, 1, 5925744000, 2323918080, 336510720, 24071040, 937440, 20160, 224, 1

Offset

1,2

Comments

Previous name was: Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...

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.

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.

Links

Table of n, a(n) for n=1..36.

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.

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

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

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

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

W. Lang, First 10 rows.

Formula

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;

E.g.f. of m-th column: ((-1+(1-7*x)^(-1/7))^m)/m!.

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.

Example

{1}; {8,1}; {120,24,1}; {2640,672,48,1}; ...

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(7*k+1, k=0..n), 8); # Peter Luschny, Jan 27 2016

Mathematica

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)!);
a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1;
Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]]
(* Jean-François Alcover, Jun 17 2011 *)
rows = 8;
a[n_, m_] := BellY[n, m, Table[Product[7k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

Crossrefs

Cf. A132060 (row sums), A132061 (alternating row sums).

Cf. A092082 S2(7) triangle.

Sequence in context: A174503 A048786 A240955 * A051187 A284865 A221758

Adjacent sequences: A132053 A132054 A132055 * A132057 A132058 A132059

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang Sep 14 2007

Extensions

New name from Peter Luschny, Jan 27 2016

Status

approved