A145662 -id:A145662

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A145660

a(n) = numerator of polynomial of genus 1 and level n for m = 4 = A[1,n](4).

+10
4

0, 4, 18, 220, 883, 17672, 23566, 659868, 5278979, 95021762, 380087174, 16723836916, 66895348819, 3478558152448, 13914232622662, 11131386100532, 178102177617521, 3027737019533893, 4036982692723202, 306810684647167556

(list; graph; refs; listen; history; text; internal format)

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_{d,1,n-1} m^(n-d)/d.

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_{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).

LINKS

Table of n, a(n) for n=1..20.

MAPLE

A145660 := proc(n) add( 4^(n-d)/d, d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

MATHEMATICA

m = 4; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

CROSSREFS

Cf. A145609-A145640, A145656, A145668, A145662, A145664, A145666.

KEYWORD

frac,nonn

AUTHOR

Artur Jasinski, Oct 16 2008

STATUS

approved

A145664

a(n) = numerator of polynomial of genus 1 and level n for m = 6 = A[1,n](6).

+10
4

0, 6, 39, 236, 2835, 42531, 255191, 10718052, 257233353, 2315100317, 2315100338, 152796622518, 1833559470601, 71508819355749, 429052916136639, 2574317496821836, 123567239847463143, 6301929232220740413

(list; graph; refs; listen; history; text; internal format)

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_{d=1..n-1} m^(n-d)/d.

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:

(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).

LINKS

Table of n, a(n) for n=1..18.

MAPLE

A145664 := proc(n) add( 6^(n-d)/d, d=1..n-1) ; numer(%) ; end proc:

seq(A145664(n), n=1..20) ; # R. J. Mathar, Feb 01 2011

MATHEMATICA

m = 6; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa

CROSSREFS

Cf. A145609-A145640, A145656, A145658, A145660, A145662, A145664.

KEYWORD

frac,nonn

AUTHOR

Artur Jasinski, Oct 16 2008

STATUS

approved

A145658

a(n) = numerator of polynomial of genus 1 and level n for m = 3

+10
3

0, 3, 21, 65, 393, 5907, 17731, 372411, 2234571, 20111419, 20111503, 663680439, 1991042087, 77650650633, 33278851497, 19967311127, 119803867191, 6109997233605, 54989975121893, 1044809527432655, 15672142912044093

(list; graph; refs; listen; history; text; internal format)

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_{d=1..n-1} m^(n-d)/d.

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:

(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).

LINKS

Table of n, a(n) for n=1..21.

MAPLE

A145658 := proc(n) add( 3^(n-d)/d, d=1..n-1) ; numer(%) ; end proc: # R. J. Mathar, Feb 01 2011

MATHEMATICA

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

CROSSREFS

Cf. A145609-A145640, A145656, A145660, A145662, A145664, A145666.

KEYWORD

frac,nonn

AUTHOR

Artur Jasinski, Oct 16 2008

STATUS

approved

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