1,2

a(n,m) := S2p(-3; n,m), a member of a sequence of triangles including S2p(-1; n,m) := A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) := A008277(n,m) (Stirling 2nd kind). a(n,1)= A008545(n-1).

a(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m increasing plane (aka ordered) trees, with one vertex of out-degree r=0 (leafs or a root) and each vertex with out-degree r>=1 comes in r+2 types (like for an (r+2)-ary vertex). Proof from the e.g.f. of the first column Y(z):=1-(1-4*x)^(1/4) and the F. Bergeron et al. reference given in A001498, eq. (8), Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w)^3. W. Lang Oct 12 2007.

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

Index entries for sequences related to Bessel functions or polynomials

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

W. Lang, First ten rows.

a(n, m) = n!*A049213(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!.

{1}; {3,1}; {21,9,1}; {231,111,18,1}; {3465,1785,345,30,1}; ...

Tree combinatorics for a(3,2)=9: there are three m=2 forests each with one tree a root

(with out-degree r=0) and the other tree a root and a leaf coming in three versions (like for a 3-ary vertex). Each such forest can be labeled increasingly in three ways (like (1,(23)), (2,(13)) and (3,(12)) yielding 9 such forests. W. Lang Oct 12 2007.

Row sums give A016036. Cf. A004747.

Columns include A008545.

Alternating row sums A132163.

Sequence in context: A024432 A016531 A107717 this_sequence A136236 A113090 A138354

Adjacent sequences: A000366 A000367 A000368 this_sequence A000370 A000371 A000372

easy,nonn,tabl

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