|
%I #13 Sep 08 2022 08:46:15
%S 0,0,0,0,10,56,224,768,2400,7040,19712,53248,139776,358400,901120,
%T 2228224,5431296,13074432,31129600,73400320,171573248,397934592,
%U 916455424,2097152000,4771020800,10796138496,24310185984,54492397568,121634816000,270448721920,599147937792
%N Expansion of (2 x^4*(5 - 12*x + 8*x^2))/(1 - 2*x)^4.
%C a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which total number of east steps below y = x-1 or above y = x+1 is exactly three. Details can be found in Section 4.1 in Pan and Remmel's link.
%H Colin Barker, <a href="/A268462/b268462.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_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F G.f.: (2 x^4*(5 - 12*x + 8*x^2))/(1 - 2*x)^4.
%F a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-3)-16*a(n-4) for n>3. - _Vincenzo Librandi_, Feb 05 2016
%F a(n) = 2^(n-4)*(n-3)*(n+1)*(n+2)/3 for n>2. - _Colin Barker_, Feb 08 2016
%t CoefficientList[Series[(2 x^4 (5 - 12 x + 8 x^2)) / (1 - 2 x)^4, {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 05 2016 *)
%t LinearRecurrence[{8,-24,32,-16},{0,0,0,0,10,56,224},40] (* _Harvey P. Dale_, Feb 10 2022 *)
%o (Magma) I:=[0,0,0,0,10,56,224]; [n le 7 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Feb 05 2016
%o (PARI) concat(vector(4), Vec(2*x^4*(5-12*x+8*x^2)/(1-2*x)^4 + O(x^100))) \\ _Colin Barker_, Feb 08 2016
%K nonn,easy
%O 0,5
%A _Ran Pan_, Feb 04 2016
|
