login
a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3n^2 + 15n + 20)/2880.
7

%I #50 Sep 02 2024 20:32:49

%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