OFFSET
1,2
COMMENTS
For the numerators of the polynomials of genus 1 and level n for m = 1 see A001008.
Definition: The polynomial A[1,2*n+1](m) = A[genus 1,level n] is here defined as Sum_{d = 1..n-1} m^(n-d)/d.
First few 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
The general formula which uses these polynomials is the following:
(1/(n+1))*Hypergeometric2F1[1, n, n+1, 1/m] = Sum_{k >= 0} m^(-k)/(k + n) = m^n * ArcTanh[(2*m-1)/(2*m^2-2*m+1)] - A[1,n](m) = (m^n)*Log[m/(m-1)] - A[1,n](m).
Conjecture: a(n) = numerator( (2^n)*log(2) - 2^(n+1)*Integral_{x = 0..1} x^(2*n-1)/(1 + x^2)^n ). - Peter Bala, Jun 10 2024
MAPLE
A145656 := proc(n) add( 2^(n-d)/d, d = 1..n-1) end: seq(numer(A145656(n)), n = 1..20); # R. J. Mathar, Feb 01 2011
MATHEMATICA
m = 2; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa
a[n_]:=2Integrate[(2-x^n)/(2-x), {x, 0, 1}]+4(2^(n-1)-1)Log[2]
Table[a[n] // Simplify // Numerator, {n, 0, 22}] (* Gerry Martens, Jun 04 2016 *)
CROSSREFS
KEYWORD
frac,nonn,easy
AUTHOR
Artur Jasinski, Oct 16 2008
STATUS
approved