%I #16 Feb 08 2016 11:05:47

%S 0,0,0,0,0,0,8,60,294,1180,4200,13776,42560,125568,357120,985600,

%T 2652672,6988800,18077696,46018560,115507200,286326784,701890560,

%U 1703411712,4096655360,9771417600,23132110848,54384394240,127049662464,295069286400,681574400000

%N Expansion of 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

%C a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly two east steps below y = x - 1 and exactly two easts step above y = x + 1. Details can be found in Section 4.1 in Pan and Remmel's link.

%H Colin Barker, <a href="/A268599/b268599.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_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).

%F G.f.: 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

%F a(n) = 2^(n-10)*(n-5)*(n-4)*(n^2+3*n+10) for n>3. - _Colin Barker_, Feb 08 2016

%t CoefficientList[Series[2 x^6 (4 - 10 x + 7 x^2)/(1 - 2 x)^5, {x, 0, 30}], x] (* _Michael De Vlieger_, Feb 08 2016 *)

%o (PARI) concat(vector(6), Vec(2*x^6*(4-10*x+7*x^2)/(1-2*x)^5 + O(x^100))) \\ _Colin Barker_, Feb 08 2016

%Y Cf. A268462, A268586, A268587, A268598.

%K nonn,easy

%O 0,7

%A _Ran Pan_, Feb 08 2016