A108648 a(n) = (n+1)^2*(n+2)^3*(n+3)/24.

1, 18, 120, 500, 1575, 4116, 9408, 19440, 37125, 66550, 113256, 184548, 289835, 441000, 652800, 943296, 1334313, 1851930, 2527000, 3395700, 4500111, 5888828, 7617600, 9750000, 12358125, 15523326, 19336968, 23901220, 29329875, 35749200

Offset: 0

Author

Emeric Deutsch, Jun 13 2005

Keywords

• nonn
• easy

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 (p. 230, no. 24).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 24).
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A108647.

Programs

Magma

[(n+1)^2*(n+2)^3*(n+3)/24: n in [0..30]]; // G. C. Greubel, Oct 28 2022

Maple

a:=(n+1)^2*(n+2)^3*(n+3)/24: seq(a(n),n=0..36);

Mathematica

Table[(n+1)^2*(n+2)^3*(n+3)/24, {n,0,30}] (* G. C. Greubel, Oct 28 2022 *)

PARI

Vec((1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Apr 22 2020

SageMath

[(n+1)^2*(n+2)^3*(n+3)/24 for n in (0..30)] # G. C. Greubel, Oct 28 2022

Formula

From Colin Barker, Apr 22 2020: (Start)

G.f.: (1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.

(End)

a(n) = A000217(n+1) * A002415(n+2). - J. M. Bergot, May 21 2022

From Amiram Eldar, May 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 24*zeta(3) + 6*Pi^2 - 87.

Sum_{n>=0} (-1)^n/a(n) = 99 - Pi^2 - 96*log(2) - 18*zeta(3). (End)

E.g.f.: (24 + 408*x + 1020*x^2 + 772*x^3 + 224*x^4 + 26*x^5 + x^6)*exp(x)/4!. - G. C. Greubel, Oct 28 2022