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a(n) = (n+1)(n+2)^3*(n+3)^3*(n+4)(2n+5)/4320.
1

%I #26 Jun 24 2020 03:08:28

%S 1,28,300,1925,8918,32928,102816,282150,698775,1591876,3383380,

%T 6782139,12931100,23609600,41505024,70570332,116486397,187250700,

%U 293916700,451511137,680159634,1006454240,1465100000,2100881250,2970992115

%N a(n) = (n+1)(n+2)^3*(n+3)^3*(n+4)(2n+5)/4320.

%C Kekulé numbers for certain benzenoids.

%C Dimensions of certain Lie algebra (see Landsberg-Manivel reference for precise definition). - _N. J. A. Sloane_, Oct 15 2007

%H Antoine Bourget, Amihay Hanany, Dominik Miketa, <a href="https://arxiv.org/abs/2005.05273">Quiver origami: discrete gauging and folding</a>, arXiv:2005.05273 [hep-th], 2020. See Eq. (3.50), 1st row in Fig. 8, and top box in Fig. 9.

%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8_13">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 229).

%H J. M. Landsberg and L. Manivel, <a href="http://dx.doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math. 201 (2006), 143-179. [Th. 7.1, case a=0]

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F G.f.: (x+1)*(x^4+17*x^3+48*x^2+17*x+1)/(x-1)^10. - _Colin Barker_, Sep 20 2012

%F a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - _Wesley Ivan Hurt_, Jun 23 2020

%p a:=n->(1/4320)*(n+1)*(n+2)^3*(n+3)^3*(n+4)*(2*n+5): seq(a(n),n=0..30);

%t Table[(1/4320)(n+1)(n+2)^3(n+3)^3(n+4)(2n+5),{n,0,30}] (* _Harvey P. Dale_, Nov 03 2011 *)

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Jun 12 2005