|
| |
|
|
A132056
|
|
Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
|
|
4
| |
|
|
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
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| 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.
|
|
|
REFERENCES
| F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, 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.
|
|
|
LINKS
| W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
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}; ...
|
|
|
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]]
(* From Jean-François Alcover, Jun 17 2011 *)
|
|
|
CROSSREFS
| Cf. A132060 (row sums), A132061 (alternating row sums).
Cf. A092082 S2(7) triangle.
Sequence in context: A114152 A174503 A048786 * A051187 A021850 A197576
Adjacent sequences: A132053 A132054 A132055 * A132057 A132058 A132059
|
|
|
KEYWORD
| nonn,easy,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 14 2007
|
| |
|
|
