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%I #15 Jun 05 2016 23:43:20
%S 0,2,5,32,131,661,1327,18608,148969,447047,89422,1967410,7869871,
%T 102309709,204620705,2046213056,32739453941,556571077357,556571247527,
%U 10574855234543,42299423848079,42299425233749,84598851790183
%N a(n) = numerator of polynomial of genus 1 and level n for m = 2
%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[m^(n - d)/d,{d,1,n-1}]
%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 following:
%C (1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =
%C Sum[m^(-x)(1/(x+n),{x,0,Infinity}] =
%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 A145656 := proc(n) add( 2^(n-d)/d,d=1..n-1) ; numer(%) ; end proc: # _R. J. Mathar_, Feb 01 2011
%t m = 2; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa
%t a[n_]:=2Integrate[(2-x^n)/(2-x),{x,0,1}]+4(2^(n-1)-1)Log[2]
%t Table[a[n] // Simplify // Numerator,{n,0,22}] (* _Gerry Martens_, Jun 04 2016 *)
%Y Cf. A145609-A145640, A145656-A145687.
%K frac,nonn
%O 1,2
%A _Artur Jasinski_, Oct 16 2008
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