A145662 - OEIS

%I #15 Jan 21 2019 04:18:50

%S 0,5,55,835,8365,41837,209195,7321885,73218955,1098284605,5491423277,

%T 302028282755,1510141416085,98159192073245,490795960391965,

%U 2453979801983849,24539798019883535,2085882831690821195

%N a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).

%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:

%C (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).

%C The sequence of denominators is ?, 1, 2, 6, 12, 12, 12, 84, ... - _Matthew J. Samuel_, Jan 30 2011

%p A145662 := proc(n) add( 5^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # _R. J. Mathar_, Feb 01 2011

%t m = 5; 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, A145658, A145660, A145664, A145666.

%Y Cf. A006245.

%K frac,nonn

%O 1,2

%A _Artur Jasinski_, Oct 16 2008