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For numerator of amazing polynomial of genus 1 and level n for m = 1 see A001008
For denominators see A145667.
Definition: Amazing polynomial A[1,n](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/3 + m^2/2 + m^3
n=5: A[1,5](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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