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%I A046089
%S A046089 1,3,1,12,9,1,60,75,18,1,360,660,255,30,1,2520,6300,3465,645,45,1,
%T A046089 20160,65520,47880,12495,1365,63,1,181440,740880,687960,235305,35700,
%U A046089 2562,84,1,1814400,9072000,10372320,4452840,877905,86940,4410,108,1
%N A046089 A triangle of numbers related to triangle A030523.
%C A046089 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).
%C A046089 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).
%C A046089 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
%C A046089 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
%D A046089 E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's 
               board, Discr. Maths. 239 (2001) 33-51.
%H A046089 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A046089 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A046089.text">
               First ten rows. </a>
%F A046089 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!.
%F A046089 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.
%Y A046089 Cf. A049376.
%Y A046089 Alternating row sums A134138.
%Y A046089 Sequence in context: A156366 A144353 A039811 this_sequence A113360 A089434 
               A019232
%Y A046089 Adjacent sequences: A046086 A046087 A046088 this_sequence A046090 A046091 
               A046092
%K A046089 easy,nonn,tabl
%O A046089 1,2
%A A046089 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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