A268586 Expansion of x^3*(3*x - 2)/(2*x - 1)^3.

0, 0, 0, 2, 9, 30, 88, 240, 624, 1568, 3840, 9216, 21760, 50688, 116736, 266240, 602112, 1351680, 3014656, 6684672, 14745600, 32374784, 70778880, 154140672, 334495744, 723517440, 1560281088, 3355443200, 7197425664, 15401484288, 32883343360, 70061654016

Offset: 0

Author

Ran Pan, Feb 07 2016

Keywords

• nonn
• easy

Comments

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have two east steps below y = x - 1 and no east steps 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 (6,-12,8).

Programs

Mathematica

CoefficientList[Series[(x^3 (3 x - 2))/(2 x - 1)^3, {x, 0, 30}], x] (* Michael De Vlieger, Feb 08 2016 *)
LinearRecurrence[{6,-12,8},{0,0,0,2,9},40] (* Harvey P. Dale, Apr 25 2020 *)

PARI

concat(vector(3), Vec(x^3*(2-3*x)/(1-2*x)^3 + O(x^100))) \\ Colin Barker, Feb 08 2016

Formula

G.f.: x^3*(3*x - 2)/(2*x - 1)^3.

From Colin Barker, Feb 08 2016: (Start)

a(n) = 2^(n-5)*(n-2)*(n+5) for n>1.

a(n) = 6*a(n-1)-12*a(n-2)+8*a(n-3) for n>4.

(End)