1,2

a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quartic (4-ary) trees. Proof based on the a(n,m) recurrence. See a D. Callan comment on the m=1 case A007559. 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.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

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

W. Lang, First 10 rows.

a(n, m) = sum(|A051141(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.

a(n, m) = n!*A035529(n, m)/(m!*3^(n-m)); a(n+1, m) = (3*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+(1-3*x)^(-1/3))^m)/m!.

{1}; {4,1}; {28,12,1}; {280,160,24,1}; {3640,2520,520,40,1}; ...

a(n, m)=: S2(4, n, m) is the fourth 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). a(n, 1)= A007559(n).

Row sums: A049119(n), n >= 1.

Sequence in context: A119304 A114150 A134149 this_sequence A073323 A077097 A123126

Adjacent sequences: A035466 A035467 A035468 this_sequence A035470 A035471 A035472

easy,nice,nonn,tabl

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