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%I A000992 M0793 N0300
%S A000992 1,1,1,2,3,6,11,24,47,103,214,481,1030,2337,5131,11813,26329,60958,
%T A000992 137821,321690,734428,1721998,3966556,9352353,21683445,51296030,
%U A000992 119663812,284198136,666132304,1586230523,3734594241,8919845275
%N A000992 a(n)= Sum_{k=1 ... floor(n/2)} a(k)a(n-k) with a(1) = 1.
%C A000992 Comment from David Callan, Nov 02 2006: a(n) = number of (unlabeled, 
               rooted) ordered trees on n-1 vertices in which all outdegrees are 
               <=2 and, for each vertex of outdegree 2, the sizes of its two subtrees 
               are weakly increasing left to right (n>=2). The number b(n) of such 
               trees on n vertices satisfies the recurrence b[1]=1; b[n_]/;n>=2 
               := b[n] = b[n-1] + Sum[b[i]b[n-1-i],{i,Floor[(n-1)/2]}], the first 
               term counting trees whose root has outdegree 1 and the sum counting 
               trees whose root has outdegree 2 by size of the left subtree. This 
               recurrence generates b(n)=a(n+1), n>=1. For example, the a(5)=3 such 
               trees are:
%C A000992 .|....|...../\..
%C A000992 .|.../.\.....|..
%C A000992 .|..............
%C A000992 Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 27 
               2009: (Start)
%C A000992 The connection with the Rayleigh polynomials Phi(2n,x) of A158616 is 
               that Phi(2n,x)= sum_{i=1..a(n)} 2^(n_i) product_{j=2..n-1} (x+j)^(n_ij), 
               as described by Kishore.
%C A000992 So a(n) counts the terms in the representation of the Polynomial Phi(2n,
               x) as a sum over these "base" polynomials.
%C A000992 For example Phi(12,x) = 2^4*(x+2)^2*(x+3) +2^2*(x+2)*(x+3)^2 +2^3*(x+2)*(x+3)*(x+4) 
               + 2^3*(x+2)*(x+3)*(x+5) +2^2*(x+2)*(x+4)*(x+5) +2*(x+3)^2*(x+5) has 
               a(6)=6 terms. (End)
%D A000992 N. Kishore, A structure of the Rayleigh polynomial, Duke Math. J., 31 
               (1964), 513-518.
%D A000992 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000992 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000992 T. D. Noe, <a href="b000992.txt">Table of n, a(n) for n=1..200</a>
%p A000992 al := 1/2; M1 := 30; a[ 0 ] := 1; for n from 0 to M1 do n0 := floor(al*n);
%p A000992 a[ n+1 ] := sum( a[ i ]*a[ n-i ], i=0..n0); i := 'i'; od: [ seq(a[ j 
               ],j=0..M1) ];
%Y A000992 Also called "1/2-Catalans", compare recurrence for A000108.
%Y A000992 A093637 counts above trees without the restriction that all outdegrees 
               are <=2.
%Y A000992 Cf. A001190, A124973.
%Y A000992 Sequence in context: A123465 A000055 A006787 this_sequence A036648 A047750 
               A072187
%Y A000992 Adjacent sequences: A000989 A000990 A000991 this_sequence A000993 A000994 
               A000995
%K A000992 nonn,easy,nice
%O A000992 1,4
%A A000992 N. J. A. Sloane (njas(AT)research.att.com).

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