0,5

Noncrossing handshakes of 2(n-1) people (each using only one hand) on round table, up to rotations - Antti Karttunen Sep 03 2000

a(n), n>2, is also the number of oriented cacti on n-1 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 (liskov(AT)im.bas-net.by), Oct 19 2005

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 285 (4.1.26), 291 (4.1.48)

A. Errara, De quelques problemes d'analysis situs, Comptes Rend. Congr. Nat. Sci. Bruxelles, (1930), 106-110.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 67, (3.3.26).

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.

G. Labelle, Sur la symetrie et l'asymetrie des structures combinatoires. Theor. Comput. Sci. 117, No. 1-2, 3-22 (1993).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. W. Walkup, The number of plane trees, Mathematika, vol. 19, No. 2 (1972), 200-204.

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for sequences related to trees

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(n-1).

a(n)=1/(2*(n-1))*sum{d|(n-1)}(phi((n-1)/d)*C(2d, d)) - A000108(n-1)/2 + (if n is even) A000108(n/2-1)/2.

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((1-sqrt(1-4*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 p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od : - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006

Cf. A000108, A003239, A005354, A057502, A061417, A064640.

Sequence in context: A032065 A099968 A010357 this_sequence A093467 A080408 A001420

Adjacent sequences: A002992 A002993 A002994 this_sequence A002996 A002997 A002998

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms, formula from Christian G. Bower (bowerc(AT)usa.net), Dec 15 1999.