

A107891


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


7



1, 19, 155, 805, 3136, 9996, 27468, 67320, 150645, 313027, 611611, 1134497, 2012920, 3436720, 5673648, 9093096, 14194881, 21643755, 32310355, 47319349, 68105576, 96479020, 134699500, 185562000, 252493605, 339663051, 452103939
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OFFSET

0,2


COMMENTS

Kekulé numbers for certain benzenoids.


REFERENCES

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).


LINKS

Paolo Aluffi, Degrees of projections of rank loci, 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]."]


FORMULA

a(n2) = (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*(ij))^2, where V(x_1,x_2} is the Vandermonde matrix of order 2.  Peter Bala, Sep 21 2007
G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1x)^9.  Colin Barker, Feb 08 2012
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


MAPLE

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);


MATHEMATICA

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 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



