A108684 a(n) = (n+1)*(n+2)*(n+3)*(19*n^3 + 111*n^2 + 200*n + 120)/720.

1, 15, 93, 372, 1141, 2926, 6594, 13476, 25509, 45397, 76791, 124488, 194649, 295036, 435268, 627096, 884697, 1224987, 1667953, 2237004, 2959341, 3866346, 4993990, 6383260, 8080605, 10138401, 12615435, 15577408, 19097457, 23256696

Offset: 0

Author

Emeric Deutsch, Jun 19 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.233, # 10).

Links

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

Programs

Maple

a:=n->(n+1)*(n+2)*(n+3)*(19*n^3+111*n^2+200*n+120)/720: seq(a(n),n=0..33);

Mathematica

Table[(n + 1) (n + 2) (n + 3) (19 n^3 + 111 n^2 + 200 n + 120)/720, {n, 0, 29}] (* or *)
CoefficientList[Series[(1 + 8 x + 9 x^2 + x^3)/(1 - x)^7, {x, 0, 29}], x] (* or *)
Table[Sum[Binomial[(n + 1 - k) + 1, 2] Apply[Subtract, Map[Binomial[# + 2, 3] &, {n + 1, k}]], {k, 0, n}], {n, 0, 29}] (* Michael De Vlieger, Jun 08 2017 *)

Formula

G.f.: (1 + 8*x + 9*x^2 + x^3)/(1-x)^7.

a(n) = Sum_{k=0...n} A000217(n+1-k) * (A000292(n+1) - A000292(k)). - J. M. Bergot, Jun 07 2017

a(n) = A050405(n) + A181888(n+1). - R. J. Mathar, Jul 22 2022