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Coefficients of x/(1+3*x+3*x^2-x^3).
5

%I #20 Aug 05 2024 08:41:45

%S 0,1,-3,6,-8,3,21,-80,180,-279,217,366,-2028,5203,-9159,9840,3160,

%T -48159,144837,-286874,377952,-128397,-1035539,3869760,-8631060,

%U 13248361,-9982143,-18429714,98483932,-250144797,436552881,-460740320,-177582480,2351521281

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

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 562.

%H Harvey P. Dale, <a href="/A108369/b108369.txt">Table of n, a(n) for n = 0..1000</a>

%H Vincent Thill, <a href="http://vincent-thill.fr/2021/04/identite-du-mois-davril-2021/">Radicaux et Ramanujan</a>, April 2021, see c(n).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-3,1).

%F x=a(n), z=a(-n-2), y=a(n)+a(n+1), t=a(-1-n)+a(-n-2) is a solution to 2*(x^3+z^3) = y^3+t^3.

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

%F a(n) = -3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(-1-n) = A108368(n).

%F a(n+1) = (-1)^n * Sum_{k=0..floor(n/3)} (-2)^k * binomial(n+2,3*k+2). - _Seiichi Manyama_, Aug 05 2024

%t CoefficientList[Series[x/(1+3x+3x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-3,-3,1},{0,1,-3},40] (* _Harvey P. Dale_, Jul 30 2024 *)

%o (PARI) {a(n)=if(n>=0, polcoeff(x/(1+3*x+3*x^2-x^3)+x*O(x^n),n), n=-1-n; polcoeff(x/(1-3*x-3*x^2-x^3)+x*O(x^n),n))}

%Y Cf. A000748, A108368.

%K sign

%O 0,3

%A _Michael Somos_, Jun 01 2005