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A072335 Expansion of 1/((1-x^2)*(1-4*x+x^2)). 5

%I

%S 1,4,16,60,225,840,3136,11704,43681,163020,608400,2270580,8473921,

%T 31625104,118026496,440480880,1643897025,6135107220,22896531856,

%U 85451020204,318907548961,1190179175640,4441809153600,16577057438760,61866420601441,230888624967004

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

%H M. R. Bremner, <a href="http://arxiv.org/abs/1303.0920">Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures</a>, arXiv preprint arXiv:1303.0920, 2013

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-4,1).

%F 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).

%F a(n)=sum{k=0..floor(n/2), U(n-2k, 2)} - _Paul Barry_, Nov 15 2003

%F 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.

%t 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 *)

%o (PARI) Vec(1/((1-x^2)*(1-4*x+x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

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

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

%Y Cf. A225682.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 15 2002

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Last modified January 23 22:16 EST 2020. Contains 331177 sequences. (Running on oeis4.)