%I #31 Aug 30 2022 09:43:24
%S 0,14,84,300,825,1925,4004,7644,13650,23100,37400,58344,88179,129675,
%T 186200,261800,361284,490314,655500,864500,1126125,1450449,1848924,
%U 2334500,2921750,3627000,4468464,5466384,6643175,8023575,9634800,11506704,13671944
%N Number of diagonal dissections of an n-gon into 4 regions.
%C Number of standard tableaux of shape (n-4,2,2,2) (n>=6). - _Emeric Deutsch_, May 20 2004
%C Number of short bushes with n+2 edges and 4 branch nodes (i.e. nodes with outdegree at least 2). A short bush is an ordered tree with no nodes of outdegree 1. Example: a(6)=14 because the only short bushes with 8 edges and 4 branch nodes are the fourteen full binary trees with 8 edges. Column 4 of A108263. - _Emeric Deutsch_, May 29 2005
%H Vincenzo Librandi, <a href="/A033276/b033276.txt">Table of n, a(n) for n = 5..1000</a>
%H David Beckwith, <a href="http://www.jstor.org/stable/2589081">Legendre polynomials and polygon dissections?</a>, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.
%H Frank R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., Vol. 204, No. 1-3 (1999), 73-112.
%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 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = binomial(n+2, 3)*binomial(n-3, 3)/4.
%F G.f.: x^6*(14-14x+6x^2-x^3)/(1-x)^7. - _Emeric Deutsch_, May 29 2005
%F From _Amiram Eldar_, Aug 30 2022: (Start)
%F Sum_{n>=6} 1/a(n) = 109/1225.
%F Sum_{n>=6} (-1)^n/a(n) = 192*log(2)/35 - 4582/1225. (End)
%t Table[(Binomial[n+2,3]Binomial[n-3,3])/4,{n,5,40}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,14,84,300,825,1925,4004},40] (* _Harvey P. Dale_, Mar 13 2014 *)
%t CoefficientList[Series[x (14 - 14 x + 6 x^2 - x^3)/(1 - x)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 15 2014 *)
%o (Magma) [(Binomial(n+2,3)*Binomial(n-3,3))/4: n in [5..50]]; // _Vincenzo Librandi_, Mar 15 2014
%Y Cf. A033275, A108263.
%K nonn,easy
%O 5,2
%A _N. J. A. Sloane_.
%E More terms from _Vincenzo Librandi_, Mar 15 2014