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%I #25 May 29 2022 03:07:14
%S 1,18,140,700,2646,8232,22176,53460,117975,242242,468468,861224,
%T 1516060,2570400,4217088,6720984,10439037,15844290,23554300,34364484,
%U 49286930,69595240,96876000,133087500,180626355,242402706,321924708,423393040
%N Kekulé numbers for certain benzenoids.
%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 3).
%H Vincenzo Librandi, <a href="/A108680/b108680.txt">Table of n, a(n) for n = 0..1000</a>
%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 G.f.: (1 + 9*x + 14*x^2 + 4*x^3)/(1 - x)^9.
%F a(n) = (n + 1)*(n + 2)^3*(n + 3)^2*(n + 4)*(n + 5)/1440 (from Maple section).
%F Sum_{n>=0} 1/a(n) = -240*zeta(3) + (400/3)*Pi^2 - 18475/18. - _Jaume Oliver Lafont_, Jul 10 2017
%F Sum_{n>=0} (-1)^n/a(n) = 4480*log(2)/3 + 180*zeta(3) - 20*Pi^2/3 - 21325/18. - _Amiram Eldar_, May 29 2022
%p a:=n->(n+1)*(n+2)^3*(n+3)^2*(n+4)*(n+5)/1440: seq(a(n),n=0..30);
%t CoefficientList[Series[(1 + 9 x + 14 x^2 + 4 x^3) / (1 - x)^9, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jul 11 2017 *)
%o (Magma) [(n+1)*(n+2)^3*(n+3)^2*(n+4)*(n+5)/1440: n in [0..30]]; // _Vincenzo Librandi_, Jul 11 2017
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 18 2005
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