This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A080420 #30 Jun 16 2025 16:08:26
%S A080420 1,7,36,162,675,2673,10206,37908,137781,492075,1732104,6022998,
%T A080420 20726199,70681653,239148450,803538792,2683245609,8910671247,
%U A080420 29443957164,96855122250,317297380491,1035574967097,3368233731366,10920608743932,35303692060125,113819103201843
%N A080420 a(n) = (n+1)*(n+6)*3^n/6.
%C A080420 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
%H A080420 Vincenzo Librandi, <a href="/A080420/b080420.txt">Table of n, a(n) for n = 0..1000</a>
%H A080420 Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal peculiar linear mean relationships in all polynomials</a>, arXiv:1706.08381 [math.GM], 2017. See p. 4.
%H A080420 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).
%F A080420 G.f.: (1-2*x)/(1-3*x)^3.
%F A080420 From _G. C. Greubel_, Dec 22 2023: (Start)
%F A080420 a(n) = (n+6)*A288834(n)/2, for n >= 1.
%F A080420 a(n) = A136158(n+2, 2).
%F A080420 E.g.f.: (1/2)*(2 + 8*x + 3*x^2)*exp(3*x). (End)
%F A080420 From _Amiram Eldar_, Jan 11 2024: (Start)
%F A080420 Sum_{n>=0} 1/a(n) = 17721/50 - 4356*log(3/2)/5.
%F A080420 Sum_{n>=0} (-1)^n/a(n) = 4392*log(4/3)/5 - 12591/50. (End)
%t A080420 CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^3, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 05 2013 *)
%t A080420 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 *)
%o A080420 (Magma) [(n+1)*(n+6)*3^n/6: n in [0..30]]; // _Vincenzo Librandi_, Aug 05 2013
%o A080420 (SageMath) [(n+1)*(n+6)*3^n/6 for n in range(31)] # _G. C. Greubel_, Dec 22 2023
%Y A080420 T(n,2) in triangle A080419.
%Y A080420 Cf. A136158, A288834.
%K A080420 easy,nonn
%O A080420 0,2
%A A080420 _Paul Barry_, Feb 19 2003