A145666 a(n) = numerator of polynomial of genus 1 and level n for m = 7 : A[1,n](7).

0, 7, 105, 2219, 31087, 1088129, 2538991, 17772957, 248821433, 15675750559, 21946050833, 1689845914645, 11828921402977, 1076431847676451, 7535022933740305, 263725802680934699, 3692161237533130831

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,n](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/3 + m^2/2 + m^3;

n=5: A[1,5](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-A145687.

Programs

Maple

A145666 := proc(n) add( 7^(n-d)/d,d=1..n-1) ; numer(%) ; end proc:
seq(A145666(n),n=1..20) ; # R. J. Mathar, Feb 01 2011

Mathematica

m = 7; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa