A072335 Expansion of 1/((1-x^2)*(1-4*x+x^2)).

1, 4, 16, 60, 225, 840, 3136, 11704, 43681, 163020, 608400, 2270580, 8473921, 31625104, 118026496, 440480880, 1643897025, 6135107220, 22896531856, 85451020204, 318907548961, 1190179175640, 4441809153600, 16577057438760, 61866420601441, 230888624967004

Offset: 0

Author

N. J. A. Sloane, Jul 15 2002

Keywords

• nonn
• easy

Links

• M. R. Bremner, Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures, arXiv preprint arXiv:1303.0920, 2013
• N. J. A. Sloane, Transforms
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (4,0,-4,1).

Crossrefs

EULER transform of A072279 (with its initial 1 omitted).

A001353(n)^2 is a bisection of a(n).

Cf. A225682.

Programs

Mathematica

CoefficientList[Series[1/((1-x^2)*(1-4x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{4,0,-4,1},{1,4,16,60},30] (* Harvey P. Dale, Aug 22 2015 *)

PARI

Vec(1/((1-x^2)*(1-4*x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (1/12)*((7-4*sqrt(3))*(2-sqrt(3))^n+(7+4*sqrt(3))*(2+sqrt(3))^n-3+(-1)^n). Recurrence: a(n) = 4*a(n-1)-4*a(n-3)+a(n-4).

a(n)=sum{k=0..floor(n/2), U(n-2k, 2)} - Paul Barry, Nov 15 2003

The g.f. can also be written as 1/(1-4*x+4*x^3-x^4), which relates this sequence to the family of sequences described in A225682.