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%I #14 Apr 22 2020 11:34:32
%S 1,29,320,2100,9898,37044,116928,323730,807675,1851421,3955952,
%T 7966322,15249780,27941200,49273344,84012300,139021461,223980645,
%U 352290400,542195192,818163038,1212563220,1767688000,2538168750,3593841615
%N a(n) = (n+1)(n+2)^3*(n+3)^2*(n+4)(5n^2 + 23n + 30)/8640.
%C Kekulé numbers for certain benzenoids.
%H Colin Barker, <a href="/A107966/b107966.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_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(0)=1, a(1)=29, a(2)=320, a(3)=2100, a(4)=9898, a(5)=37044, a(6)=116928, a(7)=323730, a(8)=807675, a(9)=1851421, 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). - _Harvey P. Dale_, Apr 18 2016
%F G.f.: (1 + 19*x + 75*x^2 + 85*x^3 + 28*x^4 + 2*x^5) / (1 - x)^10. - _Colin Barker_, Apr 22 2020
%p a:=n->(1/8640)*(n+1)*(n+2)^3*(n+3)^2*(n+4)*(5*n^2+23*n+30): seq(a(n),n=0..26);
%t Table[(n+1)(n+2)^3(n+3)^2(n+4)(5n^2+23n+30)/8640,{n,0,30}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,29,320,2100,9898,37044,116928,323730,807675,1851421},30] (* _Harvey P. Dale_, Apr 18 2016 *)
%o (PARI) Vec((1 + 19*x + 75*x^2 + 85*x^3 + 28*x^4 + 2*x^5) / (1 - x)^10 + O(x^30)) \\ _Colin Barker_, Apr 22 2020
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 12 2005