cp's OEIS Frontend

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

%I A145658 #20 Jun 02 2025 00:40:26
%S A145658 0,3,21,65,393,5907,17731,372411,2234571,20111419,20111503,663680439,
%T A145658 1991042087,77650650633,33278851497,19967311127,119803867191,
%U A145658 6109997233605,54989975121893,1044809527432655,15672142912044093
%N A145658 a(n) = numerator of polynomial of genus 1 and level n for m = 3.
%C A145658 For numerator of polynomial of genus 1 and level n for m = 1 see A001008.
%C A145658 Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
%C A145658 Sum_{d=1..n-1} m^(n-d)/d.
%C A145658 Few first A[1,n](m):
%C A145658 n=1: A[1,1](m)= 0;
%C A145658 n=2: A[1,2](m)= m;
%C A145658 n=3: A[1,3](m)= m/2 + m^2;
%C A145658 n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4.
%C A145658 General formula which uses these polynomials is:
%C A145658 (1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =
%C A145658 Sum_{x>=0} m^(-x)/(x+n) =
%C A145658 m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) =
%C A145658 m^n*log(m/(m-1)) - A[1,n](m).
%p A145658 A145658 := proc(n) add( 3^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # _R. J. Mathar_, Feb 01 2011
%t A145658 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
%Y A145658 Cf. A145609-A145640, A145656, A145660, A145662, A145664, A145666.
%K A145658 frac,nonn
%O A145658 1,2
%A A145658 _Artur Jasinski_, Oct 16 2008