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A192080 Expansion of 1/((1-x)^6 - x^6).

%I A192080 #30 Apr 11 2023 11:57:47
%S A192080 1,6,21,56,126,252,463,804,1365,2366,4368,8736,18565,40410,87381,
%T A192080 184604,379050,758100,1486675,2884776,5592405,10919090,21572460,
%U A192080 43144920,87087001,176565486,357913941,723002336,1453179126,2906358252
%N A192080 Expansion of 1/((1-x)^6 - x^6).
%H A192080 Vincenzo Librandi, <a href="/A192080/b192080.txt">Table of n, a(n) for n = 0..1000</a>
%H A192080 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6).
%F A192080 a(n) = abs(A006090(n)) = (-1)^n * A006090(n).
%F A192080 G.f.: 1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).
%F A192080 From _G. C. Greubel_, Apr 11 2023: (Start)
%F A192080 a(n) = (2^(n+5) + A010892(n) - 2*A010892(n-1) - 27*(A057083(n) - 2*A057083(n-1)))/6.
%F A192080 a(n) = (2^(n+5) + A057079(n+2) - 27*A057681(n+1))/6. (End)
%t A192080 CoefficientList[Series[1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), {x,0,50}], x] (* _Vincenzo Librandi_, Oct 15 2012 *)
%t A192080 LinearRecurrence[{6,-15,20,-15,6},{1,6,21,56,126},30] (* _Harvey P. Dale_, Feb 22 2017 *)
%o A192080 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(),m); Coefficients(R!(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))));
%o A192080 (Maxima) makelist(coeff(taylor(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), x, 0, n), x, n), n, 0, 29);
%o A192080 (PARI) Vec(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Jun 23 2011
%o A192080 (SageMath)
%o A192080 def A192080_list(prec):
%o A192080     P.<x> = PowerSeriesRing(ZZ, prec)
%o A192080     return P( 1/((1-x)^6-x^6) ).list()
%o A192080 A192080_list(51) # _G. C. Greubel_, Apr 11 2023
%Y A192080 Cf. A006090, A049016, A049017, A057079, A057083, A057681.
%Y A192080 Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), this sequence (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).
%K A192080 nonn,easy
%O A192080 0,2
%A A192080 _Bruno Berselli_, Jun 23 2011