A107891 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3n^2 + 15n + 20)/2880.

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

Offset

0,2

Comments

Kekulé numbers for certain benzenoids.

Partial sums of A114239. First differences of A047819. - Peter Bala, Sep 21 2007

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

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

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]."]

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

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

G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1-x)^9. - Colin Barker, Feb 08 2012

a(n) = (A000330(n+2)*A000538(n+2) - (A000537(n+2))^2)/4. - J. M. Bergot, Sep 17 2013

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

Cf. A005585, A006542, A047819, A114239, A114242.

Sequence in context: A022711 A355217 A254142 * A302352 A301398 A141923

Adjacent sequences: A107888 A107889 A107890 * A107892 A107893 A107894

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 12 2005

Status

approved