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%I #21 Sep 08 2022 08:45:39
%S 1,1,1,6,19,61,240,769,2869,9774,34831,121969,428640,1509301,5297641,
%T 18644406,65502139,230343541,809678160,2846468089,10006911469,
%U 35178340254,123671565271,434760784009,1528407648960,5373087522061
%N Quartic product sequence: m = 4; p = 4^3; a(n) = Product_{k=1..(n-1)/2} ( 1 + m*cos(k*Pi/n)^2 + p*cos(k*Pi/n)^4 ).
%C Ratio at n=30 is 3.515489857847214.
%H G. C. Greubel, <a href="/A152098/b152098.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,9,4,-16).
%F a(n) = Product_{k=1..(n-1)/2} ( 1 + m*cos(k*Pi/n)^2 + p*cos(k*Pi/n)^4 ), with m=4 and p=64.
%F G.f.: (1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4). - _Colin Barker_, Oct 23 2013
%p seq(coeff(series((1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Sep 26 2019
%t m=4; l=4^3; Table[Product[1 +m*Cos[k*Pi/n]^2 +l*Cos[k*Pi/n]^4, {k, (n -1)/2}], {n, 0, 30}]//Round
%t LinearRecurrence[{1,9,4,-16}, {1,1,1,6,19}, 30] (* _G. C. Greubel_, Sep 26 2019 *)
%o (PARI) a(n) = round(prod(k=1, (n-1)/2, 1+4*cos(k*Pi/n)^2+4^3*cos(k*Pi/n)^4)) \\ _Colin Barker_, Oct 23 2013
%o (PARI) my(x='x+O('x^30)); Vec((1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3 +16*x^4)) \\ _G. C. Greubel_, Sep 26 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)* (1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4) )); // _G. C. Greubel_, Sep 26 2019
%o (Sage)
%o def A152098_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4)).list()
%o A152098_list(30) # _G. C. Greubel_, Sep 26 2019
%o (GAP) a:=[1,1,6,19];; for n in [5..30] do a[n]:=a[n-1]+9*a[n-2]+4*a[n-3] -16*a[n-4]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 26 2019
%K nonn,easy
%O 0,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 24 2008
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