%I #18 May 29 2022 03:07:34
%S 1,30,350,2450,12348,49392,166320,490050,1297725,3149146,7105098,
%T 15071420,30321200,58262400,107535744,191548044,330569505,554550150,
%U 906840550,1449035742,2267198780,3479762000,5247450000,7785618750,11379460365,16402583106,23339541330
%N a(n) = binomial(n+3,2)*binomial(n+4,3)*binomial(n+5,5)/12.
%C 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. 229).
%H T. D. Noe, <a href="/A107916/b107916.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F a(n) = (1/17280)(n+1)(n+2)^3*(n+3)^3*(n+4)^2*(n+5).
%F G.f.: -(2*x^5+28*x^4+85*x^3+75*x^2+19*x+1)/(x-1)^11. - _Colin Barker_, Sep 20 2012
%F From _Amiram Eldar_, May 29 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1400*Pi^2 + 2880*zeta(3) - 51835/3.
%F Sum_{n>=0} (-1)^n/a(n) = 20*Pi^2 + 28160*log(2) + 4320*zeta(3) - 74725/3. (End)
%p a:=n->(1/12)*binomial(n+3,2)*binomial(n+4,3)*binomial(n+5,5): seq(a(n),n=0..30);
%t a[n_] := Binomial[n + 3, 2] * Binomial[n + 4, 3] * Binomial[n + 5, 5]/12; Array[a, 25, 0] (* _Amiram Eldar_, May 29 2022 *)
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 12 2005