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Expansion of (1 - x^2)^3 / ((1 - x)^3 * (1 - x^3)) in powers of x.
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%I #17 Aug 16 2021 13:40:41

%S 1,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,

%T 3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,

%U 3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3

%N Expansion of (1 - x^2)^3 / ((1 - x)^3 * (1 - x^3)) in powers of x.

%H G. C. Greubel, <a href="/A164359/b164359.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).

%F Euler transform of length 3 sequence [ 3, -3, 1].

%F Moebius transform is length 3 sequence [ 3, 0, -1].

%F a(-n) = a(n) for all n in Z. a(n+3) = a(n) unless n=0 or n=-3. a(3*n) = 2 unless n=0. a(3*n + 1) = a(3*n + 2) = 3.

%F G.f.: -1 + (1/3) * ( 8 / (1 - x) - (2 + x) / (1 + x + x^2) ).

%F a(n) = A244893(n) if n>1. - _Michael Somos_, Apr 17 2015

%e G.f. = 1 + 3*x + 3*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 2*x^9 + ...

%t a[ n_] := -Boole[n == 0] + 2 + KroneckerSymbol[ 9, n]; (* _Michael Somos_, Apr 17 2015 *)

%t CoefficientList[Series[(1-x^2)^3/((1-x)^3*(1-x^3)),{x,0,120}],x] (* or *) LinearRecurrence[{0,0,1},{1,3,3,2},120] (* or *) PadRight[{1},120,{2,3,3}] (* _Harvey P. Dale_, Aug 16 2021 *)

%o (PARI) {a(n) = -(n==0) + 2 + kronecker(9, n)};

%Y Cf. A244893.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Aug 13 2009