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A113946
Series expansion of Farey rational polynomial based on A112627.
0
1, 5, 23, 81, 367, 1297, 5871, 20753, 93935, 332049, 1502959, 5312785, 24047343, 85004561, 384757487, 1360072977, 6156119791, 21761167633, 98497916655, 348178682129, 1575966666479, 5570858914065, 25215466663663, 89133742625041
OFFSET
0,2
COMMENTS
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.
FORMULA
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).
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]
a(n) = -a(n-1)+16*a(n-2)+16*a(n-3). - Wesley Ivan Hurt, May 07 2021
MATHEMATICA
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]
CROSSREFS
Cf. A112627.
Sequence in context: A211922 A139209 A227123 * A283114 A290938 A193696
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Jan 31 2006
STATUS
approved