A268599 Expansion of 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

0, 0, 0, 0, 0, 0, 8, 60, 294, 1180, 4200, 13776, 42560, 125568, 357120, 985600, 2652672, 6988800, 18077696, 46018560, 115507200, 286326784, 701890560, 1703411712, 4096655360, 9771417600, 23132110848, 54384394240, 127049662464, 295069286400, 681574400000

Offset: 0

Author

Ran Pan, Feb 08 2016

Keywords

• nonn
• easy

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
• Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Crossrefs

Cf. A268462, A268586, A268587, A268598.

Programs

Mathematica

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 *)

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

Formula

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

a(n) = 2^(n-10)*(n-5)*(n-4)*(n^2+3*n+10) for n>3. - Colin Barker, Feb 08 2016