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A001002
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Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.
(Formerly M2852 N1146)
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6
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1, 1, 3, 10, 38, 154, 654, 2871, 12925, 59345, 276835, 1308320, 6250832, 30142360, 146510216, 717061938, 3530808798, 17478955570, 86941210950, 434299921440, 2177832612120, 10959042823020, 55322023332420, 280080119609550
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| G.f. (offset 1) is series reversion of x-x^2-x^3.
Antidiagonal sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005
a(n+1) is number of (2,3)-rooted trees on n nodes.
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 211 (3.2.73-74)
N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 395
Index entries for reversions of series
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FORMULA
| a(n)=sum(binomial(n+k, k)*binomial(k, n-k), k=ceil(n/2)..n)/(n+1); 5*n*(n+1) * a(n) = 11*n*(2*n-1) * a(n-1) + 3*(3*n-2)*(3*n-4) * a(n-2) - Len Smiley (smiley(AT)math.uaa.alaska.edu)
G.f.: (4*sin(asin((27*x+11)/16)/3)-1)/(3*x); - Paul Barry (pbarry(AT)wit.ie), Feb 02 2005
a(n) = Sum_{k=0..[n/2]} C(2*n-k, n+k)*C(n+k, k)/(n+1). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005
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EXAMPLE
| a(3)=10 because a convex pentagon can be dissected in 5 ways into triangles (draw 2 diagonals from any of the 5 vertices) and in 5 ways into a triangle and a quadrilateral (draw any of the 5 diagonals).
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MATHEMATICA
| Rest[CoefficientList[InverseSeries[Series[y - y^2 - y^3, {y, 0, 30}], x], x]]
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x-x^2-x^3+x^2*O(x^n)), n+1))
(PARI) a(n)=if(n<0, 0, sum(k=0, n\2, (2*n-k)!/k!/(n-2*k)!)/(n+1)!)
(PARI) a(n)=sum(k=0, n\2, binomial(2*n-k, n+k)*binomial(n+k, k))/(n+1) (Hanna)
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CROSSREFS
| n*a(n)=A038112(n-1), n>0.
Cf. A104978.
Sequence in context: A151060 A151061 A109085 * A151062 A000902 A151063
Adjacent sequences: A000999 A001000 A001001 * A001003 A001004 A001005
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000
Revised by Emeric Deutsch and Len Smiley, Jun 05, 2005
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