A145658 a(n) = numerator of polynomial of genus 1 and level n for m = 3.

0, 3, 21, 65, 393, 5907, 17731, 372411, 2234571, 20111419, 20111503, 663680439, 1991042087, 77650650633, 33278851497, 19967311127, 119803867191, 6109997233605, 54989975121893, 1044809527432655, 15672142912044093

Offset: 1

Author

Artur Jasinski, Oct 16 2008

Keywords

• frac
• nonn

Comments

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

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

Sum_{d=1..n-1} m^(n-d)/d.

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/4 + m^2/3 + m^3/2 + m^4.

General formula which uses these polynomials is:

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

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

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

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

Crossrefs

Cf. A145609-A145640, A145656, A145660, A145662, A145664, A145666.

Programs

Maple

A145658 := proc(n) add( 3^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

Mathematica

m = 3; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa