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%I #16 May 31 2022 03:22:44
%S 1,24,220,1225,4998,16464,46368,115830,263175,553696,1093092,2045407,
%T 3656380,6283200,10431744,16802460,26346141,40330920,60421900,
%U 88774917,128146018,182018320,254748000,351731250,479594115,646407216,861927444
%N a(n) = (n+1)*(n+2)^3*(n+3)^2*(n+4)*(3n+5)/1440.
%C Kekulé numbers for certain benzenoids.
%H Colin Barker, <a href="/A107968/b107968.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. 230).
%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 + 15*x + 40*x^2 + 25*x^3 + 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 From _Amiram Eldar_, May 31 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 7895/7 + 10935*sqrt(3)*Pi/14 + 150*Pi^2 - 98415*log(3)/14 + 720*zeta(3).
%F Sum_{n>=0} (-1)^n/a(n) = 2875/7 + 10935*sqrt(3)*Pi/7 - 105*Pi^2 - 72960*log(2)/7 - 540*zeta(3). (End)
%p a:=n->(1/1440)*(n+1)*(n+2)^3*(n+3)^2*(n+4)*(3*n+5): seq(a(n),n=0..30);
%t Table[(n + 1)*(n + 2)^3*(n + 3)^2*(n + 4)*(3 n + 5)/1440, {n, 0, 25}] (* _Amiram Eldar_, May 31 2022 *)
%o (PARI) Vec((1 + 15*x + 40*x^2 + 25*x^3 + 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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