A145656 a(n) = numerator of polynomial of genus 1 and level n for m = 2.

0, 2, 5, 32, 131, 661, 1327, 18608, 148969, 447047, 89422, 1967410, 7869871, 102309709, 204620705, 2046213056, 32739453941, 556571077357, 556571247527, 10574855234543, 42299423848079, 42299425233749, 84598851790183

Offset: 1

Author

Artur Jasinski, Oct 16 2008

Keywords

• frac
• nonn
• easy

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

a(n) appears to be a multiple of A068566(n): The sequence {a(n)/A068566(n) : n >= 2} begins [2, 1, 16, 1, 1, 1, 16, 1, 1, 2, 2, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, ...]. - Peter Bala, Aug 07 2025

Crossrefs

Cf. A145609-A145640.

Programs

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
(* or *)
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 *)