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Search: id:A145660
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| A145660 |
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a(n) = numerator of amazing polynomial of genus 1 and level n for m = 4 = A[1,n](4) |
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+0
1
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| 0, 4, 18, 220, 883, 17672, 23566, 659868, 5278979, 95021762, 380087174, 16723836916, 66895348819, 3478558152448, 13914232622662, 11131386100532, 178102177617521, 3027737019533893, 4036982692723202, 306810684647167556
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OFFSET
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1,2
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COMMENT
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For numerator of amazing polynomial of genus 1 and level n for m = 1 see A001008
For denominators see A145661.
Definition: Amazing polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
Sum[m^(n - d)/d,{d,1,n-1}]
Few first A[1,n](m):
n=1: A[1,1](m)= 0
n=2: A[1,2](m)= m
n=3: A[1,3](m)= m/2 + m^2
n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4
General formula which uses amazing polynomials is following (*Artur Jasinski*):
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =
Sum[m^(-x)(1/(x+n),{x,0,Infinity}] =
m^(n)ArcTanh[(2m-1)/(2m^2-2m+1)]-A[1,n](m) =
m^(n)Log[m/(m-1)]-A[1,n](m)
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MATHEMATICA
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m = 4; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa (*Artur Jasinski*)
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CROSSREFS
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A145609-A145640, A145656-A145687.
Sequence in context: A154731 A071173 A143993 this_sequence A156522 A086400 A082022
Adjacent sequences: A145657 A145658 A145659 this_sequence A145661 A145662 A145663
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KEYWORD
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frac,nonn
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AUTHOR
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Artur Jasinski (grafix(AT)csl.pl), Oct 16 2008
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