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%I M3515 #57 Sep 04 2023 23:37:41
%S 1,1,0,1,4,16,56,197,680,2368,8272,29162,103544,370592,1335504,
%T 4844205,17672400,64810240,238795040,883585406,3281967832,12232957152,
%U 45740929104,171529130786,644950721584,2430970600576,9183671335776,34766765428852,131873955816880
%N Number of asymmetric planted projective plane trees with n+1 nodes; bracelets (reversible necklaces) with n black beads and n-1 white beads.
%C "DHK[ n ](2n-1)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1,...
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006079/b006079.txt">Table of n, a(n) for n = 1..200</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>.
%H Z. M. Himwich and N. A. Rosenberg, <a href="https://arxiv.org/abs/1901.04465">Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees</a>, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939. (Table 1 shows twice this sequence.)
%H P. K. Stockmeyer, <a href="http://dx.doi.org/10.1007/BFb0066456">The charm bracelet problem and its applications</a>, 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.
%H P. J. Stockmeyer, <a href="/A006078/a006078.pdf">The charm bracelet problem and its applications</a>, 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]
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F 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).
%F a(n+1) = (CatalanNumber(n) - binomial(n,floor(n/2)))/2 (for n>=3). - _David Callan_, Jul 14 2006
%e For the asymmetric planted projective plane trees sequence we have a(5) = 4, a(6) = 16, a(7) = 56, ...
%t a[1] = a[2] = 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_ *)
%o (Magma) [1,1] cat [(Catalan(n) - Binomial(n, Floor(n/2)))/2: n in [2..40]]; // _Vincenzo Librandi_, Feb 16 2015
%Y Cf. A000029, A000031, A006080, A006081, A006082.
%Y Equals half the difference of A000108 and A001405.
%K nonn,nice,easy
%O 1,5
%A _N. J. A. Sloane_
%E Alternative description and more terms from _Christian G. Bower_
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