

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
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OFFSET

0,2


COMMENTS

if p[x]=k*Product[(xa(i),{i,1,n}]/Product[(xb(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(n1) + 3; generating function = 1/(exp(x)1).
a(n) = (5*(4)^n8*(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



