1,2

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. W. Lang, Sept 14 2007.

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

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.

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.

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 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.

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!.

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

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

Sequence in context: A119935 A027447 A027517 this_sequence A051186 A012034 A138324

Adjacent sequences: A092079 A092080 A092081 this_sequence A092083 A092084 A092085

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 19 2004