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 A081909 #17 Apr 24 2024 13:29:35
%S A081909 1,3,10,36,135,513,1944,7290,26973,98415,354294,1259712,4428675,
%T A081909 15411789,53144100,181752822,617003001,2080591515,6973568802,
%U A081909 23245229340,77096677311,254535261273,836828256240,2740612539186
%N A081909 a(n) = 3^n(n^2 - n + 18)/18.
%C A081909 Binomial transform of A081908. 3rd binomial transform of (1,0,1,0,0,0,...). Case k=3 where a(n,k) = k^n*(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.
%C A081909 a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used exactly twice. - _Enrique Navarrete_, Mar 29 2024
%H A081909 Vincenzo Librandi, <a href="/A081909/b081909.txt">Table of n, a(n) for n = 0..185</a>
%H A081909 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).
%F A081909 a(n) = 3^n*(n^2 - n + 18)/18.
%F A081909 G.f.: (1 - 6x + 10x^2)/(1-3x)^3.
%F A081909 E.g.f.: exp(3*x)*(1+x^2/2). - _Enrique Navarrete_, Mar 29 2024
%e A081909 a(2)=10 since the number of words of length 2 defined on {0,1,2,3} that don't use 0 or use it twice are 12, 21, 13, 31, 23, 32, 11, 22, 33, 00. - _Enrique Navarrete_, Mar 29 2024
%t A081909 Table[3^n(n^2-n+18)/18,{n,0,30}] (* or *) CoefficientList[Series[ (1-6x+10x^2)/(1-3x)^3,{x,0,30}],x] (* _Harvey P. Dale_, Apr 26 2011 *)
%o A081909 (Magma) [3^n*(n^2-n+18)/18: n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2011
%Y A081909 Cf. A006234, A081910.
%K A081909 easy,nonn
%O A081909 0,2
%A A081909 _Paul Barry_, Mar 31 2003