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 A268586 #18 Feb 21 2025 07:48:44
%S A268586 0,0,0,2,9,30,88,240,624,1568,3840,9216,21760,50688,116736,266240,
%T A268586 602112,1351680,3014656,6684672,14745600,32374784,70778880,154140672,
%U A268586 334495744,723517440,1560281088,3355443200,7197425664,15401484288,32883343360,70061654016
%N A268586 Expansion of x^3*(3*x - 2)/(2*x - 1)^3.
%C A268586 a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have two east steps below y = x - 1 and no east steps above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.
%H A268586 Colin Barker, <a href="/A268586/b268586.txt">Table of n, a(n) for n = 0..1000</a>
%H A268586 Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%H A268586 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A268586 G.f.: x^3*(3*x - 2)/(2*x - 1)^3.
%F A268586 From _Colin Barker_, Feb 08 2016: (Start)
%F A268586 a(n) = 2^(n-5)*(n-2)*(n+5) for n>1.
%F A268586 a(n) = 6*a(n-1)-12*a(n-2)+8*a(n-3) for n>4.
%F A268586 (End)
%t A268586 CoefficientList[Series[(x^3 (3 x - 2))/(2 x - 1)^3, {x, 0, 30}], x] (* _Michael De Vlieger_, Feb 08 2016 *)
%t A268586 LinearRecurrence[{6,-12,8},{0,0,0,2,9},40] (* _Harvey P. Dale_, Apr 25 2020 *)
%o A268586 (PARI) concat(vector(3), Vec(x^3*(2-3*x)/(1-2*x)^3 + O(x^100))) \\ _Colin Barker_, Feb 08 2016
%K A268586 nonn,easy
%O A268586 0,4
%A A268586 _Ran Pan_, Feb 07 2016