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 A006079 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) 5
 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, 9183671335776, 34766765428852, 131873955816880 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS "DHK[ n ](2n-1)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1... For n>2, half the number of asymmetric Dyck (n-1)-paths. E.g. the two asymmetric 3-paths are UDUUDD and UUDDUD, so a(4) = 2/2 = 1. - David Scambler, Aug 23 2012 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 1..200 C. G. Bower, Transforms (2) Z. M. Himwich, N. A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching gene trees and species trees, arxiv:1901.04465 (Table 1 shows twice 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. P. J. 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. [Scanned annotated and corrected copy] FORMULA Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (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, Jul 14 2006 EXAMPLE For the asymmetric planted projective plane trees sequence we have a(5) = 4, a(6) = 16, a(7) = 56, ... MATHEMATICA a = a = 1; a[n_] := (CatalanNumber[n-1] - Binomial[n-1, Floor[(n-1)/2]])/2; Table[ a[n], {n, 1, 26}] (* Jean-François Alcover, Mar 09 2012, after David Callan *) PROG (MAGMA) [1, 1] cat [(Catalan(n) - Binomial(n, Floor(n/2)))/2: n in [2..40]]; // Vincenzo Librandi, Feb 16 2015 CROSSREFS Cf. A000029, A000031, A006080, A006081, A006082. Equals half the difference of A000108 and A001405. Sequence in context: A057585 A255301 A097128 * A218263 A290908 A201619 Adjacent sequences:  A006076 A006077 A006078 * A006080 A006081 A006082 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Alternative description and more terms from Christian G. Bower STATUS approved

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Last modified November 12 22:16 EST 2019. Contains 329079 sequences. (Running on oeis4.)