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%I #22 May 31 2022 08:08:55
%S 1,22,190,1015,4018,12936,35784,88110,197835,412126,806806,1498861,
%T 2662660,4550560,7518624,12058236,18834453,28731990,42909790,62865187,
%U 90508726,128250760,179101000,246782250,335859615,451886526,601568982
%N a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(n^2 + 5*n + 5)/720.
%C Kekulé numbers for certain benzenoids.
%H Colin Barker, <a href="/A107959/b107959.txt">Table of n, a(n) for n = 0..1000</a>
%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 229).
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F From _Colin Barker_, Apr 22 2020: (Start)
%F G.f.: (1 + 13*x + 28*x^2 + 13*x^3 + x^4) / (1 - x)^9.
%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
%F (End)
%F Sum_{n>=0} 1/a(n) = 120*Pi^2 - 144*sqrt(5)*Pi*tan(sqrt(5)*Pi/2) - 790. - _Amiram Eldar_, May 31 2022
%p a:=n->(1/720)*(n+1)*(n+2)^2*(n+3)^2*(n+4)*(n^2+5*n+5): seq(a(n),n=0..30);
%t Table[(n+1)(n+2)^2(n+3)^2(n+4)(n^2+5n+5)/720,{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,22,190,1015,4018,12936,35784,88110,197835},30] (* _Harvey P. Dale_, Sep 27 2020 *)
%o (PARI) Vec((1 + 13*x + 28*x^2 + 13*x^3 + x^4) / (1 - x)^9 + O(x^30)) \\ _Colin Barker_, Apr 22 2020
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 12 2005
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