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%I A002056 M4941 N2115 #70 Sep 08 2022 08:44:29
%S A002056 1,14,120,825,5005,28028,148512,755820,3730650,17978180,84987760,
%T A002056 395482815,1816357725,8250123000,37119350400,165645101160,
%U A002056 733919156190,3231337461300,14147884842000,61636377252450,267325773340626,1154761882042824,4969989654817600
%N A002056 Number of diagonal dissections of a convex n-gon into n-5 regions.
%C A002056 Number of standard tableaux of shape (n-5,n-5,1,1,1) (see Stanley reference). - _Emeric Deutsch_, May 20 2004
%C A002056 Number of increasing tableaux of shape (n-2,n-2) 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 A002056 Number of noncrossing partitions of 2n-7 into n-5 blocks all of size at least 2. - _Oliver Pechenik_, May 02 2014
%D A002056 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002056 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002056 T. D. Noe, <a href="/A002056/b002056.txt">Table of n, a(n) for n=6..100</a>
%H A002056 D. Beckwith, <a href="http://www.jstor.org/stable/2589081">Legendre polynomials and polygon dissections?</a>, Amer. Math. Monthly, 105 (1998), 256-257.
%H A002056 A. Cayley, <a href="https://doi.org/10.1112/plms/s1-22.1.237">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 A002056 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 A002056 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 A002056 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 A002056 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002056 Simon Plouffe, <a href="https://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
%H A002056 R. C. Read, <a href="/A001004/a001004.pdf">On general dissections of a polygon</a>, Preprint (1974).
%H A002056 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 A002056 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 A002056 a(n) = binomial(n-3, 3)*binomial(2n-7, n-6)/(n-5).
%F A002056 G.f.: (x-1+(1-11*x+40*x^2-50*x^3+10*x^4)*(1-4*x)^(-5/2))/(2*x^5). - _Mark van Hoeij_, Oct 25 2011
%F A002056 a(n) ~ 4^n*n^(3/2)/(768*sqrt(Pi)). - _Ilya Gutkovskiy_, Apr 11 2017
%F A002056 D-finite with recurrence: -(n-1)*(n-5)*(n-6)*a(n) +2*(2*n-7)*(n-3)*(n-4)*a(n-1)=0. - _R. J. Mathar_, Feb 16 2020
%p A002056 A002056:=n->binomial(n-3,3)*binomial(2*n-7,n-6)/(n-5): seq(A002056(n), n=6..40); # _Wesley Ivan Hurt_, Apr 12 2017
%t A002056 Table[Binomial[n - 3, 3] Binomial[2n - 7, n - 6]/(n - 5), {n, 6, 50}] (* _Indranil Ghosh_, Apr 11 2017 *)
%o A002056 (Magma) [Binomial(n-3, 3)*Binomial(2*n-7, n-6)/(n-5): n in [6..30]]; // _Vincenzo Librandi_, Feb 18 2020
%K A002056 nonn
%O A002056 6,2
%A A002056 _N. J. A. Sloane_

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