0,2

G.f. (offset 1) is series reversion of x-2x^2+x^3.

Hankel transform is A005156(n+1). - Paul Barry (pbarry(AT)wit.ie), Jan 20 2007

a(n) = number of ways to connect 2n-2 points labeled 1,2,...,2n-2 in a line with 0 or more noncrossing arcs above the line such that each maximal contiguous sequence of isolated points has even length. For example, with arcs separated by dashes, a(3)=7 counts {} (no arcs), 12, 14, 23, 34, 12-34, 14-23. It does not count 13 because 2 is an isolated point. - David Callan (callan(AT)stat.wisc.edu), Sep 18 2007

Comment from Philippe Leroux (ph_ler_math(AT)yahoo.com), Oct 05 2007, In my 2003 paper I introduced L-algebras. These are K-vector spaces equipped with two binary operations > and < satisfying (x>y)<z = x>(y<z). In my arXiv paper math-ph/0709.3453 I show that the free L-algebra on one generator is related to symmetric ternary trees with odd degrees, so the dimensions of the homogenous components are 1,2,7,30,143.... These L-algebras are closely related to the so-called triplicial-algebras, 3 associative operations and 3 relations whose free object is related to even trees.

W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

E. Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.

D. E. Knuth, personal communication.

Philippe Leroux, An algebraic framework of weighted directed graphs, Int. J. Math. Math. Sci. 58. (2003).

Philippe Leroux, An equivalence of categories motivated by weighted directed graphs, arXiv:math-ph/0709.3453.

M. Noy, Enumeration of noncrossing trees on a circle, Discr. Math. 180 (1998), 301-313.

Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864.

Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496.

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

T. D. Noe, Table of n, a(n) for n=0..100

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).

I. Gessel and G. Xin, The generating function of ternary trees and continued fractions

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 432

Douglas Rogers, Comments on A1111160, A055113 and A006013

M. Somos, Number Walls in Combinatorics.

Convolution of A001764 with itself: 2*C(3n+2,n)/(3n+2), or C(3n+2,n+1)/(3n+2).

G.f.: 4/(3x)sin(1/3 arcsin(sqrt(27x/4)))^2.

BB:=[T, {T=Prod(Z, Z, F, F), F=Sequence(B), B=Prod(F, F, Z)}, unlabeled]: seq(count(BB, size=i), i=2..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007

InverseSeries[Series[y-2*y^2+y^3, {y, 0, 32}], x]

(PARI) a(n)=if(n<0, 0, (3*n+1)!/(n+1)!/(2*n+1)!)

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x-2*x^2+x^3+x^2*O(x^n)), n+1))

Cf. A121645.

Sequence in context: A027136 A116363 A046648 this_sequence A166990 A059578 A136574

Adjacent sequences: A006010 A006011 A006012 this_sequence A006014 A006015 A006016

easy,nonn,nice

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

Edited by N. J. A. Sloane (njas(AT)research.att.com), Feb 21 2008