%I #20 Jun 25 2024 08:32:09

%S 0,2,5,32,131,661,1327,18608,148969,447047,89422,1967410,7869871,

%T 102309709,204620705,2046213056,32739453941,556571077357,556571247527,

%U 10574855234543,42299423848079,42299425233749,84598851790183

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

%C For the numerators of the polynomials of genus 1 and level n for m = 1 see A001008.

%C 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.

%C First few A[1,n](m):

%C n = 1: A[1,1](m) = 0

%C n = 2: A[1,2](m) = m

%C n = 3: A[1,3](m) = m/2 + m^2

%C n = 4: A[1,4](m) = m/3 + m^2/2 + m^3

%C n = 5: A[1,5](m) = m/4 + m^2/3 + m^3/2 + m^4

%C The general formula which uses these polynomials is the following:

%C (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).

%C 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

%p 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

%t 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

%t a[n_]:=2Integrate[(2-x^n)/(2-x),{x,0,1}]+4(2^(n-1)-1)Log[2]

%t Table[a[n] // Simplify // Numerator,{n,0,22}] (* _Gerry Martens_, Jun 04 2016 *)

%Y Cf. A145609-A145640, A145656-A145687.

%K frac,nonn,easy

%O 1,2

%A _Artur Jasinski_, Oct 16 2008