1,2

a(n,1)= A001710(n+1). a(n,m)=: S1p(3; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers).

Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m) := S2(3; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+2 colors. The k roots (j=0) each come in one (or no) color. W. Lang, Oct 12 2007

a(4,2)=75=4*(3*4)+3*(3*3) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*3*4)=12 colored versions, e.g. ((1c1),(2c1,3c3,4c2)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 3 colors, c1,c2 and c3, can be chosen, and the vertex labeled 4 with j=2 can come in 4 colors, e.g. c1, c2, c3 and c4. Therefore there are 4*((1)*(1*3*4)=48 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*3)*(1*3))=27 such forests, e.g. ((1c1,3c2)(2c1,4c1)) or ((1c1,3c2)(2c1,4c2)), etc. W. Lang, Oct 12 2007

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

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

W. Lang, First ten rows.

a(n, m) = n!*A030523(n, m)/(m!*2^(n-m)); a(n, m) = (2*m+n-1)*a(n-1, m) + a(n-1, 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: ((x*(2-x)/(2*(1-x)^2))^m)/m!.

a(n, m)= sum(|S1(n, j)|* A075497(j, m), j=m..n) (matrix product), with S1(n, j) := A008275(n, j) (signed Stirling1 triangle). Priv. comm. to W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference.

Cf. A049376.

Alternating row sums A134138.

Sequence in context: A049458 A062139 A039811 this_sequence A113360 A089434 A019232

Adjacent sequences: A046086 A046087 A046088 this_sequence A046090 A046091 A046092

easy,nonn,tabl

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