1,2

If one replaces in the recurrence the '2' by '0', resp. '1', one obtains the Lah-number, resp. Stirling-number of 2nd kind, triangle A008297, resp. A008277.

The product of two lower triangular Jabotinsky matrices (see A039692 for the Knuth 1992 reference) is again such a Jabotinsky matrix: J(n,m)=sum(J1(n,j)*J2(j,m),j=m..n). The e.g.f.s of the first columns of these triangular matrices are composed in the reversed order: f(x)=f2(f1(x)). With f1(x)=-(ln(1-2*x))/2 for J1(n,m)=|A039683(n,m)| and f2(x)=exp(x)-1 for J2(n,m)=A008277(n,m) one has therefore f2(f1(x))=1/sqrt(1-2*x) - 1 = f(x) for J(n,m)=a(n,m). This proves the matrix product given below. The m-th column of a Jabotinsky matrix J(n,m) has e.g.f. (f(x)^m)/m!, m>=1.

a(n,m) gives the number of forests with m rooted ordered trees with n non-root vertices labeled in an organic way. Organic labelling means that the vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing. Proof: first for m=1 then for m>=2 using the recurrence relation for a(n,m) given below. W. Lang, Aug 07 2007.

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

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers

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(|A039683(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 comment on products of Jabotinsky matrices.

a(n, m) = n!*A035324(n, m)/(m!*2^(n-m)), n >= m >= 1; a(n+1, m)= (2*n+m)*a(n, m)+a(n, m-1); a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1.

E.g.f. of m-th column: ((x*c(x/2)/sqrt(1-2*x))^m)/m!, where c(x) = g.f. for Catalan numbers A000108.

{1}; {3,1}; {15,9,1 ]; {105,87,18,1}; {945,975,285,30,1}; ...

Combinatoric meaning of a(3,2)=9: The nine increasing path sequences for the three rooted ordered trees with leaves labeled with 1,2,3 and the root labels 0 are: {(0,3),[(0,1),(0,2)]}; {(0,3),[(0,2),(0,1)]}; {(0,3),(0,1,2)}; {(0,1),[(0,3),(0,2)]}; [(0,1),[(0,2),(0,3)]]; [(0,2),[(0,1),(0,3)]]; {(0,2),[(0,3),(0,1)]}; {(0,1),(0,2,3)}; {(0,2),(0,1,3)}.

The column sequences are A001147, A035101, A035119, ... Row sums: A049118(n), n >= 1.

Cf. A000108, A035324, A008277, A008297.

Adjacent sequences: A035339 A035340 A035341 this_sequence A035343 A035344 A035345

Sequence in context: A038553 A135896 A134144 this_sequence A039815 A134685 A130757

easy,nice,nonn,tabl

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