

A002995


Number of unlabeled planar trees (also called plane trees) with n nodes.
(Formerly M0805)


30



1, 1, 1, 1, 2, 3, 6, 14, 34, 95, 280, 854, 2694, 8714, 28640, 95640, 323396, 1105335, 3813798, 13269146, 46509358, 164107650, 582538732, 2079165208, 7457847082, 26873059986, 97239032056, 353218528324, 1287658723550, 4709785569184
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OFFSET

0,5


COMMENTS

Noncrossing handshakes of 2(n1) people (each using only one hand) on round table, up to rotations  Antti Karttunen, Sep 03 2000
Equivalently, the number of noncrossing partitions up to rotation composed of n1 blocks of size 2.  Andrew Howroyd, May 04 2018
a(n), n>2, is also the number of oriented cacti on n1 unlabeled nodes with all cutpoints of separation degree 2, i.e. ones shared only by two (cyclic) blocks. These are digraphs (without loops) that have a unique Eulerian tour. Such digraphs with labeled nodes are enumerated by A102693.  Valery A. Liskovets, Oct 19 2005
Labeled plane trees are counted by A006963.  David Callan, Aug 19 2014
This sequence is similar to A000055 but those trees are not embedded in a plane.  Michael Somos, Aug 19 2014


REFERENCES

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 304.
A. Errera, De quelques problèmes d'analysis situs, Comptes Rend. Congr. Nat. Sci. Bruxelles, (1930), 106110.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 67, (3.3.26).
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=0..200
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and TreeLike Structures, Cambridge, 1998, pp. 285 (4.1.26), 291 (4.1.48)
CombOS  Combinatorial Object Server, Generate free plane trees
R. Cori, M. Marcus, Counting nonisomorphic chord diagrams, Theor. Comp. Sci. 204 (1998) 5575, Corollary 5.2.
Paul Drube and Puttipong Pongtanapaisan, Annular NonCrossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
A. Errera, Reviews of two articles on Analysis Situs, from Fortschritte [Annotated scanned copy]
D. Feldman, Counting plane trees, Unpublished manuscript, 1992. (Annotated scanned copy)
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322335. (Annotated scanned copy)
G. Labelle, Sur la symétrie et l'asymétrie des structures combinatoires, Theor. Comput. Sci. 117, No. 12, 322 (1993).
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 5380, 1992. (Annotated scanned copy)
T. Mütze, Proof of the middle levels conjecture, arXiv preprint arXiv:1404.4442 [math.CO], 2014.
Torsten Mütze and Franziska Weber, Construction of 2factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.
T. Mütze and F. Weber, Construction of 2factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 18321855.
J. Sawada, Generating rooted and free plane trees, ACM Transactions on Algorithms, Vol. 2 No. 1 (2006), 113.
Seunghyun Seo and Heesung Shin, Two involutions on vertices of ordered trees, FPSAC'02 (2002). (see p_n).
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209233. See Table 1.
D. W. Walkup, The number of plane trees, Mathematika, vol. 19, No. 2 (1972), 200204.
Index entries for sequences related to trees


FORMULA

G.f.: 1+B(x)+(C(x^2)C(x)^2)/2 where B is g.f. of A003239 and C is g.f. of A000108(n1).
a(n) = 1/(2*(n1))*sum{d(n1)}(phi((n1)/d)*binomial(2d, d))  A000108(n1)/2 + (if n is even) A000108(n/21)/2.


EXAMPLE

G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 14*x^7 + 34*x^8 + 95*x^9 + ...
a(7) = 14 = 11 + 3 because there are 11 trees with 7 nodes but three of them can be embedded in a plane in two ways. These three trees have degree sequences 4221111, 3321111, 3222111, where there are two trees with each degree sequence but in the first, the two nodes of degree two are adjacent, in the second, the two nodes of degree three are adjacent, and in the third, the node of degree three is adjacent to two nodes of degree two.  Michael Somos, Aug 19 2014


MAPLE

with (powseries): with (combstruct): n := 27: Order := n+2: sys := {C = Cycle(B), B = Union(Z, Prod(B, B))}: G003239 := (convert(gfseries(sys, unlabeled, x) [C(x)], polynom)) / x: G000108 := convert(taylor((1sqrt(14*x)) / (2*x), x), polynom): G002995 := 1 + G003239 + (eval(G000108, x=x^2)  G000108^2)/2: A002995 := 1, 1, 1, seq(coeff(G002995, x^i), i=1..n); # Ulrich Schimke, Apr 05 2002
with(combinat): with(numtheory): m := 2: for p from 2 to 28 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p1 do if gcd(m, p1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)binomial(m*p, p)/(p*(m1)+1)) od : # Zerinvary Lajos, Dec 01 2006


MATHEMATICA

a[0] = a[1] = 1; a[n_] := (1/(2*(n1)))*Sum[ EulerPhi[(n1)/d]*Binomial[2*d, d], {d, Divisors[n1]}]  CatalanNumber[n1]/2 + If[ EvenQ[n], CatalanNumber[n/21]/2, 0]; Table[ a[n], {n, 0, 29}] (* JeanFrançois Alcover, Mar 07 2012, from formula *)


PROG

(PARI) catalan(n) = binomial(2*n, n)/(n+1);
a(n) = if (n<2, 1, n; sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/(2*n)  catalan(n)/2 + if ((n1) % 2, 0, catalan((n1)/2)/2)); \\ Michel Marcus, Jan 23 2016


CROSSREFS

Column k=2 of A303694 and A303864.
Cf. A000055, A000108, A002996, A003239, A005354, A057502, A061417, A064640.
Sequence in context: A010357 A190166 A238823 * A093467 A246640 A080408
Adjacent sequences: A002992 A002993 A002994 * A002996 A002997 A002998


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms, formula from Christian G. Bower, Dec 15 1999
Name corrected ("labeled" > "unlabeled") by David Callan, Aug 19 2014


STATUS

approved



