1,2

a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quintic (5-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.

E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.

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) = n!*A048882(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*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+(1-4*x)^(-1/4))^m)/m!.

a(n, m) = sum(|A051142(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to 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.

{1}; {5,1}; {45,15,1}; {585,255,30,1}; {9945,5175,825,50,1}; ...

a(n, m) := S2(5, n, m) is the fifth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m), S2(4, n, m) := A035469(n, m). a(n, 1)= A007696(n). A007559(n).

Cf. A048882, A007696. Row sums: A049120(n), n >= 1.

Sequence in context: A114154 A134273 A048897 this_sequence A051150 A144341 A144342

Adjacent sequences: A049026 A049027 A049028 this_sequence A049030 A049031 A049032

easy,nonn,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)