A268462 Expansion of (2 x^4*(5 - 12*x + 8*x^2))/(1 - 2*x)^4.

0, 0, 0, 0, 10, 56, 224, 768, 2400, 7040, 19712, 53248, 139776, 358400, 901120, 2228224, 5431296, 13074432, 31129600, 73400320, 171573248, 397934592, 916455424, 2097152000, 4771020800, 10796138496, 24310185984, 54492397568, 121634816000, 270448721920, 599147937792

Offset: 0

Author

Ran Pan, Feb 04 2016

Keywords

• nonn
• easy

Comments

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.

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 (8,-24,32,-16).

Programs

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

Mathematica

CoefficientList[Series[(2 x^4 (5 - 12 x + 8 x^2)) / (1 - 2 x)^4, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 05 2016 *)
LinearRecurrence[{8,-24,32,-16},{0,0,0,0,10,56,224},40] (* Harvey P. Dale, Feb 10 2022 *)

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

Formula

G.f.: (2 x^4*(5 - 12*x + 8*x^2))/(1 - 2*x)^4.

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

a(n) = 2^(n-4)*(n-3)*(n+1)*(n+2)/3 for n>2. - Colin Barker, Feb 08 2016