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Number of diagonal dissections of a convex (n+8)-gon into n+1 regions.
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%I #35 Feb 10 2020 06:27:15

%S 1,27,385,4004,34398,259896,1790712,11511720,70114902,409003595,

%T 2303105805,12593413560,67173369900,350777861280,1798432526880,

%U 9073909567440,45140379405030,221768094898350,1077403874372826,5182007298602904,24699073588138180,116759256962107760

%N Number of diagonal dissections of a convex (n+8)-gon into n+1 regions.

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

%C Number of increasing tableaux of shape (n+6,n+6) with largest entry 2n+7. 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 Number of noncrossing partitions of 2n+7 into n+1 blocks all of size at least 2. - _Oliver Pechenik_, May 02 2014

%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 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. 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+5, 5)*binomial(2n+7, n)/(n+1).

%F G.f.: 3F2(4,6,9/2 ; 2,8 ; 4*x). - _R. J. Mathar_, Feb 09 2020

%F D-finite with recurrence n*(n+7)*(n+1)*a(n) -2*(n+5)*(n+3)*(2*n+7)*a(n-1)=0. - _R. J. Mathar_, Feb 09 2020

%t Table[(Binomial[n+5,5]Binomial[2n+7,n])/(n+1),{n,0,30}] (* _Harvey P. Dale_, Oct 16 2016 *)

%o (PARI) vector(30, n, n--; binomial(n+5, 5)*binomial(2*n+7, n)/(n+1)) \\ _Michel Marcus_, Jun 18 2015

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Michel Marcus_, Jun 18 2015