A145656 a(n) = numerator of polynomial of genus 1 and level n for m = 2

0, 2, 5, 32, 131, 661, 1327, 18608, 148969, 447047, 89422, 1967410, 7869871, 102309709, 204620705, 2046213056, 32739453941, 556571077357, 556571247527, 10574855234543, 42299423848079, 42299425233749, 84598851790183

Offset

1,2

Comments

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

Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as

Sum[m^(n - d)/d,{d,1,n-1}]

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/4 + m^2/3 + m^3/2 + m^4

General formula which uses these polynomials is following:

(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =

Sum[m^(-x)(1/(x+n),{x,0,Infinity}] =

m^(n)ArcTanh[(2m-1)/(2m^2-2m+1)]-A[1,n](m) =

m^(n)Log[m/(m-1)]-A[1,n](m)

Links

Table of n, a(n) for n=1..23.

Maple

A145656 := proc(n) add( 2^(n-d)/d, d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

Mathematica

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
a[n_]:=2Integrate[(2-x^n)/(2-x), {x, 0, 1}]+4(2^(n-1)-1)Log[2]
Table[a[n] // Simplify // Numerator, {n, 0, 22}] (* Gerry Martens, Jun 04 2016 *)

Crossrefs

Cf. A145609-A145640, A145656-A145687.

Sequence in context: A041397 A042811 A261045 * A221680 A009274 A019036

Adjacent sequences: A145653 A145654 A145655 * A145657 A145658 A145659

Keyword

frac,nonn

Author

Artur Jasinski, Oct 16 2008

Status

approved