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 A072335 Expansion of 1/((1-x^2)*(1-4*x+x^2)). 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 linear recurrences with constant coefficients, signature (4,0,-4,1). 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. 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 *) PROG (PARI) Vec(1/((1-x^2)*(1-4*x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012 CROSSREFS EULER transform of A072279 (with its initial 1 omitted). A001353(n)^2 is a bisection of a(n). Cf. A225682. Sequence in context: A269673 A231896 A128650 * A081161 A032106 A269462 Adjacent sequences:  A072332 A072333 A072334 * A072336 A072337 A072338 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jul 15 2002 STATUS approved

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Last modified July 21 02:02 EDT 2019. Contains 325189 sequences. (Running on oeis4.)