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A114014 A series expansion of a quadric Jasinski rational polynomial based on A112627 that is Farey like in p[1/2]=1. 0
1, 7, 33, 127, 529, 2031, 8465, 32495, 135441, 519919, 2167057, 8318703, 34672913, 133099247, 554766609, 2129587951, 8876265745, 34073407215, 142020251921, 545174515439, 2272324030737, 8722792247023, 36357184491793 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

if p[x]=k*Product[(x-a(i),{i,1,n}]/Product[(x-b(i),{i,1,n}] and a(i), b(i) and k are real and rational, then p[x] will be rational. If you keep the roots out of the interior of the interval (0,1), then Farey function on that interval will be relatively smooth as well. In this case the 1/4 root makes that impossible but adding an x for a zero roots ties down the zero end and the other end is at 2.

LINKS

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (-1,16,16).

FORMULA

b(n) = coefficients of (9/64)*(x + 1/2)^4/(x*(x - 1/4)*(x + 1/4)*(x + 1)); a(n) = 2*a(n-1) + 3; generating function = 1/(exp(x)-1).

a(n) = (5*(-4)^n-8*(-1)^n+243*4^n)/120 for n>1. G.f.: -(16*x^4 +32*x^3 +24*x^2 +8*x +1) / ((x +1)*(4*x -1)*(4*x +1)). [Colin Barker, Dec 03 2012]

MATHEMATICA

b = Delete[ -(64/9)*ReplacePart[Table[Coefficient[Series[(9/64)*(x + 1/2)^4/((x - 1/4)*(x + 1/4)*(x + 1)), {x, 0, 30}], x^n], {n, -1, 30}], -9/64, 2], 1]

CROSSREFS

Cf. A112627.

Sequence in context: A221036 A000605 A215054 * A229515 A320907 A258458

Adjacent sequences:  A114011 A114012 A114013 * A114015 A114016 A114017

KEYWORD

nonn,uned,easy

AUTHOR

Roger L. Bagula, Jan 31 2006

STATUS

approved

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Last modified September 29 04:22 EDT 2020. Contains 337420 sequences. (Running on oeis4.)