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

0,2

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 (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).

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

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 *)

Prog

(PARI) Vec((1 + 11*x + 15*x^2 + 3*x^3) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Apr 22 2020
(Magma) [(n+1)^2*(n+2)^3*(n+3)/24: n in [0..30]]; // G. C. Greubel, Oct 28 2022
(SageMath) [(n+1)^2*(n+2)^3*(n+3)/24 for n in (0..30)] # G. C. Greubel, Oct 28 2022

Crossrefs

Cf. A108647.

Sequence in context: A044350 A044731 A190705 * A264360 A223046 A037064

Adjacent sequences: A108645 A108646 A108647 * A108649 A108650 A108651

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 13 2005

Status

approved