A145666 - OEIS

%I #10 Jan 20 2019 23:20:22

%S 0,7,105,2219,31087,1088129,2538991,17772957,248821433,15675750559,

%T 21946050833,1689845914645,11828921402977,1076431847676451,

%U 7535022933740305,263725802680934699,3692161237533130831

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

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

%C Definition: The polynomial A[1,n](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/3 + m^2/2 + m^3;

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

%C General formula which uses these polynomials is:

%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 A145666 := proc(n) add( 7^(n-d)/d,d=1..n-1) ; numer(%) ; end proc:

%p seq(A145666(n),n=1..20) ; # _R. J. Mathar_, Feb 01 2011

%t 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

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

%K frac,nonn

%O 1,2

%A _Artur Jasinski_, Oct 16 2008