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A006079
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Number of asymmetric planted projective plane trees with n+1 nodes; bracelets (reversible necklaces) with n black beads and n-1 white beads.
(Formerly M3515)
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3
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1, 1, 0, 1, 4, 16, 56, 197, 680, 2368, 8272, 29162, 103544, 370592, 1335504, 4844205, 17672400, 64810240, 238795040, 883585406, 3281967832, 12232957152, 45740929104, 171529130786, 644950721584, 2430970600576
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| "DHK[ n ](2n-1)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1...
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to bracelets
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
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FORMULA
| Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalans (A000108), let d(x) = x/(1-x-x^2*c(x^2)) = g.f. for A001405. Then g.f. for the asymmetric planted projective plane trees sequence is (x*c(x)-d(x))/2 (the initial terms from this version are slightly different).
a(n+1) = (CatalanNumber(n)-binomial(n,Floor[n/2]))/2 (for n>=3). - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006
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EXAMPLE
| For the asymmetric planted projective plane trees sequence we have a(5) = 4, a(6) = 16, a(7) = 56, ...
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CROSSREFS
| Cf. A000029, A000031, A006080-A006082.
Sequence in context: A025182 A057585 A097128 * A201619 A197532 A122032
Adjacent sequences: A006076 A006077 A006078 * A006080 A006081 A006082
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Alternative description and more terms from Christian G. Bower (bowerc(AT)usa.net).
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