Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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