A006079 Number of asymmetric planted projective plane trees with n+1 nodes; bracelets (reversible necklaces) with n black beads and n-1 white beads.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

			For the asymmetric planted projective plane trees sequence we have a(5) = 4, a(6) = 16, a(7) = 56, ...
		

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 and N. A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939. (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]
• Index entries for sequences related to bracelets
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000029, A000031, A006080, A006081, A006082.

Equals half the difference of A000108 and A001405.

Programs

Magma

[1,1] cat [(Catalan(n) - Binomial(n, Floor(n/2)))/2: n in [2..40]]; // Vincenzo Librandi, Feb 16 2015

Mathematica

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 *)

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

Extensions

Alternative description and more terms from Christian G. Bower