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%I #16 Apr 25 2020 15:05:20
%S 0,0,0,2,9,30,88,240,624,1568,3840,9216,21760,50688,116736,266240,
%T 602112,1351680,3014656,6684672,14745600,32374784,70778880,154140672,
%U 334495744,723517440,1560281088,3355443200,7197425664,15401484288,32883343360
%N Expansion of (x^3*(3*x - 2))/(2*x - 1)^3.
%C 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 Colin Barker, <a href="/A268586/b268586.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F G.f.: (x^3*(3*x - 2))/(2*x - 1)^3.
%F From _Colin Barker_, Feb 08 2016: (Start)
%F a(n) = 2^(n-5)*(n-2)*(n+5) for n>1.
%F a(n) = 6*a(n-1)-12*a(n-2)+8*a(n-3) for n>4.
%F (End)
%t CoefficientList[Series[(x^3 (3 x - 2))/(2 x - 1)^3, {x, 0, 30}], x] (* _Michael De Vlieger_, Feb 08 2016 *)
%t LinearRecurrence[{6,-12,8},{0,0,0,2,9},40] (* _Harvey P. Dale_, Apr 25 2020 *)
%o (PARI) concat(vector(3), Vec(x^3*(2-3*x)/(1-2*x)^3 + O(x^100))) \\ _Colin Barker_, Feb 08 2016
%K nonn,easy
%O 0,4
%A _Ran Pan_, Feb 07 2016
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