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%I #51 Feb 22 2024 02:25:00
%S 0,5,21,56,120,225,385,616,936,1365,1925,2640,3536,4641,5985,7600,
%T 9520,11781,14421,17480,21000,25025,29601,34776,40600,47125,54405,
%U 62496,71456,81345,92225,104160,117216,131461,146965,163800,182040,201761,223041,245960
%N Number of diagonal dissections of an n-gon into 3 regions.
%C Number of standard tableaux of shape (n-3,2,2) (n>=5). - _Emeric Deutsch_, May 13 2004
%C Number of short bushes with n+1 edges and 3 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(5)=5 because the only short bushes with 6 edges and 3 branch nodes are the five full binary trees with 6 edges. Column 3 of A108263. - _Emeric Deutsch_, May 29 2005
%H Indranil Ghosh, <a href="/A033275/b033275.txt">Table of n, a(n) for n = 4..10000</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), pp. 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), pp. 73-112.
%H Ronald C. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. Math., Vol. 18 (1978), pp. 370-388, Table 1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = binomial(n+1, 2)*binomial(n-3, 2)/3.
%F G.f.: z^5*(5-4*z+z^2)/(1-z)^5. - _Emeric Deutsch_, May 29 2005
%F From _Amiram Eldar_, Jan 06 2021: (Start)
%F Sum_{n>=5} 1/a(n) = 43/150.
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
%F E.g.f.: x*(exp(x)*(12 - 6*x + x^3) - 6*(2 + x))/12. - _Stefano Spezia_, Feb 21 2024
%t a[4]=0; a[n_]:=Binomial[n+1,2]*Binomial[n-3,2]/3; Table[a[n],{n,4,43}] (* _Indranil Ghosh_, Feb 20 2017 *)
%o (PARI) concat(0, Vec(z^5*(5-4*z+z^2)/(1-z)^5 + O(z^60))) \\ _Michel Marcus_, Jun 18 2015
%o (PARI) a(n) = binomial(n+1, 2)*binomial(n-3, 2)/3 \\ _Charles R Greathouse IV_, Feb 20 2017
%o (Sage)
%o def A033275(n): return (binomial(n+1, 2)*binomial(n-3, 2))//3
%o print([A033275(n) for n in range(4,50)]) # _Peter Luschny_, Apr 03 2020
%Y 2nd skew subdiagonal of A033282.
%Y Cf. A033276, A108263.
%K nonn,easy
%O 4,2
%A _N. J. A. Sloane_
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