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A006013
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a(n) = C(3*n+1,n)/(n+1).
(Formerly M1782)
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50
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1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 23841480, 142498692, 859515920, 5225264024, 31983672534, 196947587823, 1219199353190, 7583142491925, 47365474641870, 296983176369495, 1868545312633440, 11793499763070480
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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G.f. (offset 1) is series reversion of x-2x^2+x^3.
Hankel transform is A005156(n+1). - Paul Barry, 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, Sep 18 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 homogeneous 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. - Philippe Leroux (ph_ler_math(AT)yahoo.com), Oct 05 2007
a(n) is also the number of projective dependency trees with n nodes. [From Marco Kuhlmann (marco.kuhlmann(AT)lingfil.uu.se), Apr 06 2010]
Number of subpartitions of [1^2,2^2,...,n^2]. - Franklin T. Adams-Watters, Apr 13 2011.
a(n) = sum of (n+1)-th row terms of triangle A143603. - Gary W. Adamson, Jul 07 2011
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REFERENCES
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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.
S. Kitaev and A. de Mier, Enumeration of fixed points of an involution on beta(1, 0)-trees, arXiv preprint arXiv:1210.2618, 2012. - From N. J. A. Sloane, Dec 31 2012
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).
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LINKS
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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 A111160, A055113 and A006013
M. Somos, Number Walls in Combinatorics.
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FORMULA
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Convolution of A001764 with itself: 2*C(3*n+2,n)/(3*n+2), or C(3*n+2,n+1)/(3*n+2).
G.f.: 4/(3x)sin(1/3 arcsin(sqrt(27x/4)))^2.
G.f.: A(x)/x with A(x)=x/(1-A(x))^2 [Vladimir Kruchinin, Dec 26 2010]
a(n) is the top left term in M^n, where M is the infinite square production matrix:
2, 1, 0, 0, 0,...
3, 2, 1, 0, 0,...
4, 3, 2, 1, 0,...
5, 4, 3, 2, 1,...
...
- Gary W. Adamson, Jul 14, 2011
a(n) is the sum of top row terms in Q^n, where Q is the infinite square production matrix as follows:
1, 1, 0, 0, 0,...
2, 2, 1, 0, 0,...
3, 3, 2, 1, 0,...
4, 4, 3, 2, 1,...
... - Gary W. Adamson, Aug 11 2011
2*(n+1)*(2n+1)*a(n) = 3*(3n-1)(3n+1)*a(n-1). - R. J. Mathar, Dec 17 2011
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EXAMPLE
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a(3) = 30 since the top row of Q^3 = (12, 12, 5, 1)
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MAPLE
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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
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MATHEMATICA
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InverseSeries[Series[y-2*y^2+y^3, {y, 0, 32}], x]
Binomial[3#+1, #]/(#+1)&/@Range[0, 30] (* Harvey P. Dale, Mar 16 2011 *)
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PROG
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(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))
(Sage)
def A006013_list(n) :
D = [0]*(n+1); D[1] = 1
R = []; b = false; h = 1
for i in range(2*n) :
for k in (1..h) : D[k] += D[k-1]
if b : R.append(D[h]); h += 1
b = not b
return R
A006013_list(23) # Peter Luschny, May 03 2012
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CROSSREFS
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Cf. A121645, A115728.
Cf. A143603
Sequence in context: A186858 A174796 A046648 * A187979 A196148 A193464
Adjacent sequences: A006010 A006011 A006012 * A006014 A006015 A006016
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KEYWORD
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easy,nonn,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by N. J. A. Sloane, Feb 21 2008
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STATUS
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approved
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