A114244 a(n) = (n+1)*(n+2)^2*(n+3)*(7n^2 + 28n + 30)/360.

1, 13, 76, 295, 889, 2254, 5040, 10242, 19305, 34243, 57772, 93457, 145873, 220780, 325312, 468180, 659889, 912969, 1242220, 1664971, 2201353, 2874586, 3711280, 4741750, 6000345, 7525791, 9361548, 11556181, 14163745, 17244184, 20863744, 25095400, 30019297

Offset: 0

Author

Emeric Deutsch, Nov 18 2005

Keywords

• nonn
• easy

Comments

Kekulé numbers for certain benzenoids.

First differences of A114242. - 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 (pp. 167-169, Table 10.5/II/5).

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A114242.

Programs

Maple

a:=n->(n+1)*(n+2)^2*(n+3)*(7*n^2+28*n+30)/360: seq(a(n),n=0..35);

Mathematica

Table[(n + 1)*(n + 2)^2*(n + 3)*(7*n^2 + 28*n + 30)/360, {n, 0, 30}] (* Amiram Eldar, May 31 2022 *)
CoefficientList[Series[(1+x)(1+5x+x^2)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{1,13,76,295,889,2254,5040},40] (* Harvey P. Dale, Mar 06 2023 *)

Formula

G.f.: (1+x)(1 + 5x + x^2)/(1-x)^7.

From Amiram Eldar, May 31 2022: (Start)

Sum_{n>=0} 1/a(n) = 5*Pi*(7*sqrt(14)*coth(sqrt(2/7)*Pi) - 6*Pi) - 1295/9.

Sum_{n>=0} (-1)^n/a(n) = 5*Pi*(7*sqrt(14)*cosech(sqrt(2/7)*Pi) + 3*Pi) - 2755/9. (End)