A080420 a(n) = (n+1)*(n+6)*3^n/6.

1, 7, 36, 162, 675, 2673, 10206, 37908, 137781, 492075, 1732104, 6022998, 20726199, 70681653, 239148450, 803538792, 2683245609, 8910671247, 29443957164, 96855122250, 317297380491, 1035574967097, 3368233731366, 10920608743932, 35303692060125, 113819103201843

Offset: 0

Author

Paul Barry, Feb 19 2003

Keywords

• easy
• nonn

Comments

a(n-1) is the number of words of length n defined on 5 letters that have exactly one a and no b's or exactly two b's and no a's. For example, for n=3, a(2) = 36 since the words are (number of permutations in parentheses): acc (3), add (3), aee (3), acd (6), ace (6), ade (6), bbc (3), bbd (3), bbe (3). - Enrique Navarrete, Jun 10 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See p. 4.
• Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Crossrefs

T(n,2) in triangle A080419.

Cf. A136158, A288834.

Programs

Magma

[(n+1)*(n+6)*3^n/6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

Mathematica

CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)
Table[(n+1)(n+6)3^n/6,{n,0,30}] (* or *) LinearRecurrence[{9,-27,27},{1,7,36},30] (* Harvey P. Dale, Apr 02 2019 *)

SageMath

[(n+1)*(n+6)*3^n/6 for n in range(31)] # G. C. Greubel, Dec 22 2023

Formula

G.f.: (1-2*x)/(1-3*x)^3.

From G. C. Greubel, Dec 22 2023: (Start)

a(n) = (n+6)*A288834(n)/2, for n >= 1.

a(n) = A136158(n+2, 2).

E.g.f.: (1/2)*(2 + 8*x + 3*x^2)*exp(3*x). (End)

From Amiram Eldar, Jan 11 2024: (Start)

Sum_{n>=0} 1/a(n) = 17721/50 - 4356*log(3/2)/5.

Sum_{n>=0} (-1)^n/a(n) = 4392*log(4/3)/5 - 12591/50. (End)