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%I #16 Apr 21 2023 02:05:39
%S 1,3,8,18,37,75,152,309,631,1290,2636,5385,10999,22464,45881,93711,
%T 191404,390942,798497,1630923,3331144,6803829,13896755,28383990,
%U 57974032,118411413,241854191,493984896,1008959473,2060790171
%N A Chebyshev transform of A090400 related to the knot 8_2.
%C The g.f. is a transformation of the g.f. 1/(1-3*x+3*x^3) of A090400 under the Chebyshev transform G(x) -> (1/(1+x^2))*G(x/(1+x^2)). The denominator of the g.f. is a parameterization of the Alexander polynomial for the knot 8_2.
%H G. C. Greubel, <a href="/A099845/b099845.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-3,3,-1).
%F G.f.: (1 + x^2)^2/(1 - 3*x + 3*x^2 - 3*x^3 + 3*x^4 - 3*x^5 + x^6).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A090400(n-2*k).
%F a(n) = Sum_{k=0..n} A099844(n-k)*binomial(2, k/2)*(1+(-1)^k)/2.
%F a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6). - _Wesley Ivan Hurt_, Oct 10 2021
%t CoefficientList[Series[(1+x^2)^2/(1-3*x+3*x^2-3*x^3+3*x^4-3*x^5+x^6), {x,0,40}], x] (* _Wesley Ivan Hurt_, Oct 10 2021 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^2)^2/(1-3*x+3*x^2-3*x^3+3*x^4-3*x^5+x^6) )); // _G. C. Greubel_, Apr 20 2023
%o (SageMath)
%o def A077952_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x^2)^2/(1-3*x+3*x^2-3*x^3+3*x^4-3*x^5+x^6) ).list()
%o A077952_list(40) # _G. C. Greubel_, Apr 20 2023
%Y Cf. A090400, A099844.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 27 2004
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