%I A001002 M2852 N1146
%S A001002 1,1,3,10,38,154,654,2871,12925,59345,276835,1308320,6250832,30142360,
%T A001002 146510216,717061938,3530808798,17478955570,86941210950,434299921440,
%U A001002 2177832612120,10959042823020,55322023332420,280080119609550
%N A001002 Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals 
               by nonintersecting diagonals.
%C A001002 G.f. (offset 1) is series reversion of x-x^2-x^3.
%C A001002 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
%C A001002 a(n+1) is number of (2,3)-rooted trees on n nodes.
%D A001002 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001002 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001002 N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and 
               related issues, Discr. Math., 308 (2008), 1209-1221.
%D A001002 I. M. H. Etherington, Some problems of non-associative combinations, 
               Edinburgh Math. Notes, 32 (1940), 1-6.
%D A001002 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.
%D A001002 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 
               Structures, Cambridge, 1998, p. 211 (3.2.73-74)
%H A001002 T. D. Noe, <a href="b001002.txt">Table of n, a(n) for n=0..100</a>
%H A001002 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=395">
               Encyclopedia of Combinatorial Structures 395</a>
%H A001002 <a href="Sindx_Res.html#revert">Index entries for reversions of series</
               a>
%F A001002 a(n)=sum(binomial(n+k, k)*binomial(k, n-k), k=ceil(n/2)..n)/(n+1); 5n(n+1) 
               * a(n) = 11n(2n-1) * a(n-1) + 3(3n-2)(3n-4) * a(n-2) - Len Smiley 
               (smiley(AT)math.uaa.alaska.edu)
%F A001002 G.f.: (4sin(asin((27x+11)/16)/3)-1)/(3x); - Paul Barry (pbarry(AT)wit.ie), 
               Feb 02 2005
%F A001002 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
%e A001002 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).
%t A001002 Rest[CoefficientList[InverseSeries[Series[y - y^2 - y^3, {y, 0, 30}], 
               x], x]]
%o A001002 (PARI) a(n)=if(n<0,0,polcoeff(serreverse(x-x^2-x^3+x^2*O(x^n)),n+1))
%o A001002 (PARI) a(n)=if(n<0,0,sum(k=0,n\2,(2*n-k)!/k!/(n-2*k)!)/(n+1)!)
%o A001002 (PARI) a(n)=sum(k=0,n\2,binomial(2*n-k,n+k)*binomial(n+k,k))/(n+1) (Hanna)
%Y A001002 n*a(n)=A038112(n-1), n>0.
%Y A001002 Cf. A104978.
%Y A001002 Sequence in context: A151060 A151061 A109085 this_sequence A151062 A000902 
               A151063
%Y A001002 Adjacent sequences: A000999 A001000 A001001 this_sequence A001003 A001004 
               A001005
%K A001002 nonn,easy,nice
%O A001002 0,3
%A A001002 N. J. A. Sloane (njas(AT)research.att.com).
%E A001002 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000
%E A001002 Revised by Emeric Deutsch and Len Smiley, Jun 05, 2005