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%I #47 May 29 2022 03:07:42
%S 1,19,155,805,3136,9996,27468,67320,150645,313027,611611,1134497,
%T 2012920,3436720,5673648,9093096,14194881,21643755,32310355,47319349,
%U 68105576,96479020,134699500,185562000,252493605,339663051,452103939
%N a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3n^2 + 15n + 20)/2880.
%C Kekulé numbers for certain benzenoids.
%C Partial sums of A114239. First differences of A047819. - _Peter Bala_, Sep 21 2007
%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 167, 187 and p. 105 eq. (iii) for k=2 and m=5).
%H Shawn A. Broyles, <a href="/A107891/b107891.txt">Table of n, a(n) for n = 0..1000</a>
%H Paolo Aluffi, <a href="https://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
%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 a(n-2) = (1/8) * Sum_{1 <= x_1, x_2 <= n} (x_1*x_2)^2*(det V(x_1,x_2))^2 = 1/8*sum {1 <= i,j <= n} (i*j*(i-j))^2, where V(x_1,x_2} is the Vandermonde matrix of order 2. - _Peter Bala_, Sep 21 2007
%F G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1-x)^9. - _Colin Barker_, Feb 08 2012
%F a(n) = (A000330(n+2)*A000538(n+2) - (A000537(n+2))^2)/4. - _J. M. Bergot_, Sep 17 2013
%F Sum_{n>=0} 1/a(n) = 17095/4 - 240*Pi^2 - 162*sqrt(15)*Pi*tanh(sqrt(5/3)*Pi/2). - _Amiram Eldar_, May 29 2022
%p a:=n->(1/2880)*(n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n^2+15*n+20): seq(a(n),n=0..32);
%t Table[((1+n) (2+n)^2 (3+n)^2 (4+n) (20+3 n (5+n)))/2880,{n,0,40}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,19,155,805,3136,9996,27468,67320,150645},40] (* _Harvey P. Dale_, Dec 10 2021 *)
%Y Cf. A005585, A006542, A047819, A114239, A114242.
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jun 12 2005
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