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 A123050 Conjectured number of ordered trees on n edges for which the conjugate and transpose commute. 6
 1, 1, 2, 2, 2, 4, 4, 4, 6, 6, 6, 8, 8, 8, 12, 12, 12, 14, 16, 16, 18, 18, 22, 24, 24, 24, 30, 30, 30, 32, 38, 38, 40, 40, 46, 48, 48, 48, 58, 58, 58, 60, 68, 68, 70, 70, 80, 82, 82, 82, 94, 94, 94, 96, 108, 108, 110, 110, 122, 124, 124, 124, 140, 140, 140, 142, 156, 156, 158 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The conjugate of an ordered tree is given by flipping it over, while its transpose is given by flipping over the corresponding binary tree. A list of ordered trees for which the conjugate and transpose commute, counted by this sequence, is given in Exercise 17, Sec. 7.2.1.6 of the Knuth reference. (Knuth deletes the root from an ordered tree and works with the resulting forest instead.) This list is complete provided a certain set of ordered trees contains no self-conjugate members other than the "obvious" ones. The set in question consists of all trees generated by repeatedly applying the following two productions to the one-edge tree: (i) T -> plant(T) (i.e. add an edge to the root to obtain a new root) and (ii) T -> add left root edge to the transpose of the conjugate of T. Computational evidence suggests that this proviso does indeed hold. REFERENCES D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley Professional; 1ST edition (Feb 06, 2006). LINKS D. E. Knuth, Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6. FORMULA a(0)=a(1)=1, a(n) = 2(Floor[(n+1)/3] + Sum[Max[0,Floor[(n-(8k+2))/4]],{k,1,(n-2)/8}]) for n>=2. GF: 1 + x + 2x^2/((1-x)(1-x^3)) + 2x^14/((1-x)*(1-x^4)*(1-x^8)) MATHEMATICA a[0]=a[1]=1; a[n_]/; n>=2 := 2(Floor[(n+1)/3] + Sum[Max[0, Floor[(n-(8k+2))/4]], {k, 1, (n-2)/8}]); Table[a[n], {n, 0, 90}] CROSSREFS This sequence updates the lower bound conjectured in A079438. Sequence in context: A086227 A302402 A079438 * A113694 A086159 A029048 Adjacent sequences:  A123047 A123048 A123049 * A123051 A123052 A123053 KEYWORD nonn AUTHOR David Callan, Sep 25 2006 STATUS approved

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Last modified May 19 06:57 EDT 2019. Contains 323386 sequences. (Running on oeis4.)