

A006079


Number of asymmetric planted projective plane trees with n+1 nodes; bracelets (reversible necklaces) with n black beads and n1 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
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OFFSET

1,5


COMMENTS

"DHK[ n ](2n1)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1...
For n>2, half the number of asymmetric Dyck (n1)paths. E.g. the two asymmetric 3paths 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 nonmatching 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. 339349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. SpringerVerlag, 1974.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. SpringerVerlag, 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


FORMULA

Let c(x) = (1sqrt(14*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = x/(1xx^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[1] = a[2] = 1; a[n_] := (CatalanNumber[n1]  Binomial[n1, Floor[(n1)/2]])/2; Table[ a[n], {n, 1, 26}] (* JeanFrançois 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

N. J. A. Sloane


EXTENSIONS

Alternative description and more terms from Christian G. Bower


STATUS

approved



