STATUS
proposed
approved
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Revision History for A145662
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).
(history; published version)
(history; published version)
STATUS
editing
proposed
COMMENTS
General formula which uses these polynomials is:
STATUS
editing
proposed
NAME
a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).
COMMENTS
Sum[_{d=1..n-1} m^(n - d)/d,{d,1,n-1}].
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.
(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] = Sum[_{x>=0} m^(-x)(1/(x+n),{x,0,Infinity}] = m^(n)ArcTanh[*arctanh((2m-1)/(2m^2-2m+1)]) - A[1,n](m) = m^(n)Log[*log(m/(m-1)]) - A[1,n](m).
The Sequence sequence of denominators is ?, 1, 2, 6, 12, 12, 12, 84, ... - _Matthew J. Samuel, _, Jan 30 2011
MAPLE
A145662 := proc(n) add( 5^(n-d)/d, d=1..n-1) ; numer(%) ; end proc: # _R. J. Mathar, _, Feb 01 2011
MATHEMATICA
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 (*Artur Jasinski*)
STATUS
approved
editing
COMMENTS
Definition: Amazing The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as
NAME
a(n) = numerator of amazing polynomial of genus 1 and level n for m = 5 = A[1,n](5)
COMMENTS
For numerator of amazing polynomial of genus 1 and level n for m = 1 see A001008
COMMENTS
General formula which uses amazing these polynomials is:
AUTHOR
_Artur Jasinski (grafix(AT)csl.pl), _, Oct 16 2008