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%I #20 Apr 21 2023 09:23:37
%S 1,4,0,-320,-6400,-102400,-1536000,-22528000,-327680000,-4751360000,
%T -68812800000,-996147200000,-14417920000000,-208666624000000,
%U -3019898880000000,-43704647680000000,-632501043200000000,-9153649049600000000,-132472897536000000000
%N Expansion of (1-16*x)/(1-20*x+80*x^2).
%C Tenth binomial transform of A099839. Fifteenth binomial transform of A099840.
%H G. C. Greubel, <a href="/A099841/b099841.txt">Table of n, a(n) for n = 0..850</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-80).
%F G.f.: (1-16*x)/(1-20*x+80*x^2).
%F a(n) = (10 - 2*sqrt(5))^n*(1/2 + 3*sqrt(5)/10) + (10 + 2*sqrt(5))^n*(1/2 - 3*sqrt(5)/10).
%F a(n) = 20*a(n-1) - 80*a(n-2); a(0)=1, a(1)=4. - _Harvey P. Dale_, Jun 04 2013
%F a(n) = 2^(2*n-1)*5^((n-1)/2)*(-sqrt(5)*(1+(-1)^n)*Fibonacci(n-2) - (1 - (-1)^n)*Lucas(n-2)). - _G. C. Greubel_, Apr 21 2023
%t CoefficientList[Series[(1-16x)/(1-20x+80x^2),{x,0,30}],x] (* or *) LinearRecurrence[{20,-80},{1,4},30] (* _Harvey P. Dale_, Jun 04 2013 *)
%o (Magma) [n le 2 select 3*n-2 else 20*(Self(n-1) - 4*Self(n-2)): n in [1..41]]; // _G. C. Greubel_, Apr 21 2023
%o (SageMath)
%o def A099841(n): return 4^n*5^((n-1)/2)*(-sqrt(5)*((n-1)%2)*fibonacci(n-2) - (n%2)*lucas_number2(n-2,1,-1))
%o [A099841(n) for n in range(41)] # _G. C. Greubel_, Apr 21 2023
%Y Cf. A000032, A000045, A099839, A099840.
%K easy,sign
%O 0,2
%A _Paul Barry_, Oct 27 2004
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