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A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.
(Formerly M4639 N1982)

%I M4639 N1982

%S 1,9,56,300,1485,7007,32032,143208,629850,2735810,11767536,50220040,

%T 212952285,898198875,3771484800,15775723920,65770848990,273420862110,

%U 1133802618000,4691140763400,19371432850770,79850555673174

%N Number of diagonal dissections of a convex n-gon into n-4 regions.

%C Number of standard tableaux of shape (n-4,n-4,1,1) (see Stanley reference). - _Emeric Deutsch_, May 20 2004

%C Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-6. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - _Oliver Pechenik_, May 02 2014

%C a(n) = number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - _Oliver Pechenik_, May 02 2014

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A002055/b002055.txt">Table of n, a(n) for n = 5..100</a>

%H D. Beckwith, <a href="http://www.jstor.org/stable/2589081">Legendre polynomials and polygon dissections?</a>, Amer. Math. Monthly, 105 (1998), 256-257.

%H A. Cayley, <a href="http://plms.oxfordjournals.org/content/s1-22/1/237.extract">On the partitions of a polygon</a>, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

%H P. Lisonek, <a href="http://dx.doi.org/10.1006/jsco.1995.1066">Closed forms for the number of polygon dissections</a>, Journal of Symbolic Computation 20 (1995), 595-601.

%H O. Pechenik, <a href="http://arxiv.org/abs/1209.1355">Cyclic sieving of increasing tableaux and small Schröder paths</a>, arXiv:1209.1355 [math.CO], 2012-2014.

%H O. Pechenik, <a href="http://dx.doi.org/10.1016/j.jcta.2014.04.002">Cyclic sieving of increasing tableaux and small Schröder paths</a>, J. Combin. Theory A, 125 (2014), 357-378.

%H R. C. Read, <a href="/A001004/a001004.pdf">On general dissections of a polygon</a>, Preprint (1974)

%H Ronald C. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. math. 18 (1978) 370-388, Table 1.

%H R. P. Stanley, <a href="http://dx.doi.org/10.1006/jcta.1996.0099">Polygon dissections and standard Young tableaux</a>, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

%F a(n) = binomial(n-3, 2)*binomial(2*n-6, n-5)/(n-4).

%F With offset 0, this has a(n)=(n+2)*C(2n+4,n)/2 and e.g.f. dif(dif(x*dif(exp(2x)*Bessel_I(2,2x),x),x),x)/2. - _Paul Barry_, Aug 25 2007

%F G.f.: 16*x^5*(x+sqrt(1-4x))/((1-4x)^(3/2) *(1+sqrt(1-4x))^4 ). - _R. J. Mathar_, Nov 17 2011

%F D-finite with recurrence: (n-1)*a(n) +(23-11n)*a(n-1) +10*(4n-13)*a(n-2) +10*(23-5n)*a(n-3) +4*(2n-13)*a(n-4)=0. - _R. J. Mathar_, Nov 17 2011

%F a(n) ~ 4^n*sqrt(n)/(128*sqrt(Pi)). - _Ilya Gutkovskiy_, Apr 11 2017

%t Table[(Binomial[n-3,2]Binomial[2n-6,n-5])/(n-4),{n,5,30}] (* _Harvey P. Dale_, Nov 06 2011 *)

%o (PARI) a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);

%o for(n=5, 30, print1(a(n),", ")) \\ _Indranil Ghosh_, Apr 11 2017

%Y a(n) = f(n,n+1) where f is given in A034261.

%K nonn,nice,easy

%O 5,2

%A _N. J. A. Sloane_

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Last modified April 18 14:46 EDT 2021. Contains 343089 sequences. (Running on oeis4.)