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%I M5094 #43 Aug 14 2022 16:53:44
%S 1,20,225,1925,14014,91728,556920,3197700,17587350,93486536,483367885,
%T 2442687975,12109051500,59053512000,283963030560,1348824395160,
%U 6338392712550,29503515951000,136173391604250
%N Number of diagonal dissections of a convex (n+6)-gon into n regions.
%C Number of standard tableaux of shape (n,n,1,1,1,1) (see Stanley reference). - _Emeric Deutsch_, May 20 2004
%C Number of increasing tableaux of shape (n+4,n+4) with largest entry 2n+4. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, such that the set of entries forms an initial segment of the positive integers. - _Oliver Pechenik_, May 02 2014
%C a(n) = number of noncrossing partitions of 2n+4 into n blocks all of size at least 2. - _Oliver Pechenik_, May 02 2014
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007160/b007160.txt">Table of n, a(n) for n = 1..150</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 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].
%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 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 D-finite with recurrence (n+5)(n-1)*n*a(n) = 2(2n+3)(n+3)(n+2)a(n-1).
%F a(n) = binomial(n+3, 4)*binomial(2n+4, n-1)/n.
%t a[n_] := (n+1)(n+2)(n+3)*Binomial[2n+4, n-1]/24; Table[a[n], {n, 1, 19}](* _Jean-François Alcover_, Nov 16 2011 *)
%o (Magma) [Binomial(n+3, 4)*Binomial(2*n+4, n-1)/n : n in [1..30]]; // _Vincenzo Librandi_, Nov 17 2011
%Y A diagonal of A033282.
%K easy,nonn,nice
%O 1,2
%A _N. J. A. Sloane_
%E Offset is correct!
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