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A145658
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a(n) = numerator of amazing polynomial of genus 1 and level n for m = 3
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1
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0, 3, 21, 65, 393, 5907, 17731, 372411, 2234571, 20111419, 20111503, 663680439, 1991042087, 77650650633, 33278851497, 19967311127, 119803867191, 6109997233605, 54989975121893, 1044809527432655, 15672142912044093
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OFFSET
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1,2
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COMMENTS
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For numerator of amazing polynomial of genus 1 and level n for m = 1 see A001008.
Definition: Amazing 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 amazing polynomials is:
(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)
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LINKS
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Table of n, a(n) for n=1..21.
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MAPLE
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A145658 := proc(n) add( 3^(n-d)/d, d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011
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MATHEMATICA
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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
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CROSSREFS
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A145609-A145640, A145656-A145687.
Sequence in context: A129212 A117984 A050615 * A188667 A083564 A054646
Adjacent sequences: A145655 A145656 A145657 * A145659 A145660 A145661
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KEYWORD
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frac,nonn
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AUTHOR
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Artur Jasinski, Oct 16 2008
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STATUS
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approved
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