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Expansion of (1-x+2x^2-2x^3)/(1-x+x^2)^2.
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%I #14 Nov 06 2016 03:36:40

%S 1,1,1,-1,-4,-4,1,7,7,-1,-10,-10,1,13,13,-1,-16,-16,1,19,19,-1,-22,

%T -22,1,25,25,-1,-28,-28,1,31,31,-1,-34,-34,1,37,37,-1,-40,-40,1,43,43,

%U -1,-46,-46,1,49,49,-1,-52,-52,1,55,55,-1,-58,-58,1,61,61,-1

%N Expansion of (1-x+2x^2-2x^3)/(1-x+x^2)^2.

%C a(n+1) is the Hankel transform of {1,0,1,3,9,28,90,297,1001,3432,11934,...}, cf. A000245.

%C Binomial transform of A128214.

%C a(n+2) is the Hankel transform of A014138. - _Paul Barry_, Mar 15 2008

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,2,-1).

%F a(n) = cos(Pi*n/3) + (2n/sqrt(3)-1/sqrt(3))*sin(Pi*n/3).

%F a(n) = y(n,n), where y(m+1,n) = y(m,n) - y(m-1,n), with y(0,n)=1 and y(1,n)=n. - _Benedict W. J. Irwin_, Nov 05 2016

%t Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == y[m] - y[m - 1], y[0] == 1, y[1] == n}]][n], {n, 0, 100}] (* _Benedict W. J. Irwin_, Nov 05 2016 *)

%o (PARI) Vec((1-x+2*x^2-2*x^3)/(1-x+x^2)^2 + O(x^100)) \\ _Michel Marcus_, May 31 2014

%K easy,sign

%O 0,5

%A _Paul Barry_, Feb 19 2007