Search: id:A035342
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%I A035342
%S A035342 1,3,1,15,9,1,105,87,18,1,945,975,285,30,1,10395,12645,4680,705,45,1,
%T A035342 135135,187425,82845,15960,1470,63,1,2027025,3133935,1595790,370125,
%U A035342 43890,2730,84,1,34459425,58437855,33453945,8998290
%N A035342 A triangle of numbers related to the triangle A035324; generalization 
               of Stirling numbers of second kind A008277 and Lah numbers A008297.
%C A035342 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.
%C A035342 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.
%C A035342 a(n,m) gives the number of forests with m rooted ordered trees with n 
               non-root vertices labeled in an organic way. Organic labeling 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.
%D A035342 E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's 
               board, Discr. Maths. 239 (2001) 33-51.
%H A035342 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/
               abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized 
               Bell Numbers</a>
%H A035342 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 A035342 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/
               abs/quant-ph/0402027">The general boson normal ordering problem.</
               a>
%H A035342 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A035342.text">
               First 10 rows</a>.
%F A035342 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.
%F A035342 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.
%F A035342 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.
%e A035342 {1}; {3,1}; {15,9,1 ]; {105,87,18,1}; {945,975,285,30,1}; ...
%e A035342 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)}.
%Y A035342 The column sequences are A001147, A035101, A035119, ... Row sums: A049118(n), 
               n >= 1.
%Y A035342 Cf. A000108, A035324, A008277, A008297.
%Y A035342 Sequence in context: A038553 A135896 A134144 this_sequence A039815 A147453 
               A147020
%Y A035342 Adjacent sequences: A035339 A035340 A035341 this_sequence A035343 A035344 
               A035345
%K A035342 easy,nice,nonn,tabl
%O A035342 1,2
%A A035342 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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