login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = numerator of polynomial of genus 1 and level n for m = 4 = A[1,n](4).
4

%I #13 Jan 21 2019 04:18:40

%S 0,4,18,220,883,17672,23566,659868,5278979,95021762,380087174,

%T 16723836916,66895348819,3478558152448,13914232622662,11131386100532,

%U 178102177617521,3027737019533893,4036982692723202,306810684647167556

%N a(n) = numerator of polynomial of genus 1 and level n for m = 4 = A[1,n](4).

%C For numerator of polynomial of genus 1 and level n for m = 1 see A001008.

%C Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as

%C Sum_{d,1,n-1} m^(n-d)/d.

%C Few first 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/4 + m^2/3 + m^3/2 + m^4.

%C General formula which uses these polynomials is following:

%C (1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =

%C Sum_{x>=0} m^(-x)/(x+n) =

%C m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) =

%C m^n*log(m/(m-1)) - A[1,n](m).

%p A145660 := proc(n) add( 4^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # _R. J. Mathar_, Feb 01 2011

%t 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

%Y Cf. A145609-A145640, A145656, A145668, A145662, A145664, A145666.

%K frac,nonn

%O 1,2

%A _Artur Jasinski_, Oct 16 2008