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%I #11 May 07 2021 14:00:17
%S 1,5,23,81,367,1297,5871,20753,93935,332049,1502959,5312785,24047343,
%T 85004561,384757487,1360072977,6156119791,21761167633,98497916655,
%U 348178682129,1575966666479,5570858914065,25215466663663,89133742625041
%N Series expansion of Farey rational polynomial based on A112627.
%C Polynomial expanded is constant*(x+1/2)^2*(1+2x)/(1-x-16x^2+16x^3) the Jasinski rational polynomial p[x_] = (9/32)*(x + 1/2)^3/((x - 1/4)*(x + 1/4)*(x + 1)) f[x_] := 1/p[x] /; 0 <= x <= 1/2 f[x_] := p[x] /; 1/2 < x <= 1 gives a Farey like function with maximum at 1.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,16,16).
%F b(n) = coefficient series expansion of (9/32)*(x + 1/2)^3/((x - 1/4)*(x + 1/4)*(x + 1)) a(n) = (-16/9)*b(n).
%F a(n) = (5*(-4)^n+4*(-1)^n+81*4^n)/60 for n>0. G.f.: -(2*x+1)^3 / ((x+1)*(4*x-1)*(4*x+1)). [_Colin Barker_, Dec 03 2012]
%F a(n) = -a(n-1)+16*a(n-2)+16*a(n-3). - _Wesley Ivan Hurt_, May 07 2021
%t b = -(16/9)*ReplacePart[Table[Coefficient[Series[(9/32)*(x + 1/2)^3/((x - 1/4)*(x + 1/4)*(x + 1)), {x, 0, 30}], x^n], {n, 0, 30}], -9/16, 1]
%Y Cf. A112627.
%K nonn,easy
%O 0,2
%A _Roger L. Bagula_, Jan 31 2006