cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A145660 a(n) = numerator of polynomial of genus 1 and level n for m = 4 = A[1,n](4).

Original entry on oeis.org

0, 4, 18, 220, 883, 17672, 23566, 659868, 5278979, 95021762, 380087174, 16723836916, 66895348819, 3478558152448, 13914232622662, 11131386100532, 178102177617521, 3027737019533893, 4036982692723202, 306810684647167556
Offset: 1

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Author

Artur Jasinski, Oct 16 2008

Keywords

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).

Crossrefs

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

Programs

  • 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

A145662 a(n) = numerator of polynomial of genus 1 and level n for m = 5 = A[1,n](5).

Original entry on oeis.org

0, 5, 55, 835, 8365, 41837, 209195, 7321885, 73218955, 1098284605, 5491423277, 302028282755, 1510141416085, 98159192073245, 490795960391965, 2453979801983849, 24539798019883535, 2085882831690821195
Offset: 1

Views

Author

Artur Jasinski, Oct 16 2008

Keywords

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).
The sequence of denominators is ?, 1, 2, 6, 12, 12, 12, 84, ... - Matthew J. Samuel, Jan 30 2011

Crossrefs

Cf. A145609-A145640, A145656, A145658, A145660, A145664, A145666.
Cf. A006245.

Programs

  • 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

A145658 a(n) = numerator of polynomial of genus 1 and level n for m = 3.

Original entry on oeis.org

0, 3, 21, 65, 393, 5907, 17731, 372411, 2234571, 20111419, 20111503, 663680439, 1991042087, 77650650633, 33278851497, 19967311127, 119803867191, 6109997233605, 54989975121893, 1044809527432655, 15672142912044093
Offset: 1

Views

Author

Artur Jasinski, Oct 16 2008

Keywords

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).

Crossrefs

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

Programs

  • 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
Showing 1-3 of 3 results.

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