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.

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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

A145610 Denominator of the polynomial A_l(x) = Sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=1.

Original entry on oeis.org

2, 12, 20, 280, 2520, 27720, 360360, 720720, 4084080, 15519504, 5173168, 356948592, 8923714800, 80313433200, 2329089562800, 144403552893600, 13127595717600, 13127595717600, 485721041551200, 485721041551200
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For numerators and explicit examples of the polynomials see A145609.

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    A := proc(l,x) add(x^(l-d)/d,d=1..l-1) ; end: A145610 := proc(n) denom( A(2*n+1,1)) ; end: seq(A145610(n),n=1..20) ; # R. J. Mathar, Aug 21 2009
  • Mathematica
    m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Denominator[k]], {r, 1, 30}]; aa (* Artur Jasinski *)

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145612 Denominator of the polynomial A_l(x) = Sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=2.

Original entry on oeis.org

1, 6, 15, 420, 63, 1386, 9009, 360360, 1531530, 29099070, 14549535, 1338557220, 1673196525, 10039179150, 145568097675, 72201776446800, 18050444111700, 9025222055850, 166966608033225, 667866432132900, 6845630929362225
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For numerators see A145611. For general properties of A_l(x) see A145609.

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    A := proc(l,x) add(x^(l-d)/d,d=1..l-1) ; end: A145612 := proc(n) denom( A(2*n+1,2)) ; end: seq(A145612(n),n=1..20) ; # R. J. Mathar, Aug 21 2009
  • Mathematica
    m = 2; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Denominator[k]], {r, 1, 30}]; aa (* Artur Jasinski *)

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145613 Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=3.

Original entry on oeis.org

21, 393, 17731, 2234571, 20111503, 1991042087, 33278851497, 119803867191, 54989975121893, 15672142912044093, 987345003473390379, 204380415719298965303, 9197118707369867504211, 248322205098990353297597
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For denominators see A145614. For general properties of A_l(x) see A145609.

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    A := proc(l,x) add(x^(l-d)/d,d=1..l-1) ; end: A145613 := proc(n) numer( A(2*n+1,3)) ; end: seq(A145613(n),n=1..20) ; # R. J. Mathar, Aug 21 2009
  • Mathematica
    m = 3; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (* Artur Jasinski *)
a[n_,m_]:=Integrate[(m-x^n)/(m-x),{x,0,1}]+(m^n-m)Log[m/(m-1)]
Table[3 a[2 n, 3] //FullSimplify //Numerator, {n,1,10}]  (* Gerry Martens , Jun 04 2016 *)

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145621 Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=7.

Original entry on oeis.org

105, 31087, 2538991, 248821433, 21946050833, 11828921402977, 7535022933740305, 3692161237533130831, 1025190103621701235981, 954451986471803883166747, 15589382445706130101521201
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For denominators see A145622. For general properties of A_l(x) see A145609.

Links

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    f:= n -> numer(add(7^(2*n+1-d)/d, d=1..2*n)):
map(f, [$1..40]); # Robert Israel, Jun 05 2016
  • Mathematica
    m = 7; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (* Artur Jasinski *)
a[n_,m_]:=Integrate[(m-x^n)/(m-x),{x,0,1}]+(m^n-m)Log[m/(m-1)]
Table[7 a[2 n, 7] // FullSimplify  // Numerator, {n,1,25}]  (* Gerry Martens , Jun 04 2016 *)

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145623 Numerator of the polynomial A_l(x) = Sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=8.

Original entry on oeis.org

68, 13126, 4200532, 1881839401, 361313167484, 254364469931206, 211631238983010892, 5417759717965164721, 2947261286573050252868, 17919348622364145592266214, 1146838311831305317954669876
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For denominators see A145624. For general properties of A_l(x) see A145609.

Links

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    G:= (32*sqrt(x)*ln((1-sqrt(x))/(1+sqrt(x))) + 4*ln(1-x))/(64*x-1):
S:= series(G, x, 51):
seq(coeff(S,x,n),n=1..50); # Robert Israel, Mar 09 2016
  • Mathematica
    m = 8; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (* Artur Jasinski, Oct 14 2008 *)
a[n_,m_]:=Integrate[(m-x^n)/(m-x),{x,0,1}]+(m^n-m)Log[m/(m-1)]
Table[8 a[2 n, 8] // Simplify  // Numerator, {n,1,25}]  (* Gerry Martens , Jun 04 2016 *)

Formula

Sum_{n >= 1} (a(n)/A145624(n))*x^n = (32*sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) - 4*log(1-x))/(1-64*x). - Robert Israel, Mar 09 2016

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145624 Denominator of the polynomial A_l(x) = Sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=8.

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 18018, 153153, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 18050444111700, 265447707525, 265447707525, 9821565178425, 9821565178425, 57526310330775
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For numerators see A145623. For general properties of A_l(x) see A145609.

Links

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    G:= (32*sqrt(x)*ln((1-sqrt(x))/(1+sqrt(x))) + 4*ln(1-x))/(64*x-1):
S:=series(G,x,101):
seq(denom(coeff(S,x,n)),n=1..100); # Robert Israel, Mar 09 2016
  • Mathematica
    m = 8; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Denominator[k]], {r, 1, 30}]; aa (* Artur Jasinski, Oct 14 2008 *)

Formula

Sum_{n >= 1} (A145623(n)/a(n))*x^n = (32*sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) - 4*log(1-x))/(1-64*x). - Robert Israel, Mar 09 2016

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145629 Denominator of the polynomial A_l(x) = Sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=10.

Original entry on oeis.org

1, 6, 1, 28, 63, 1386, 1287, 10296, 14586, 277134, 323323, 89237148, 22309287, 401567166, 5822723907, 2888071057872, 722017764468, 361008882234, 6678664321329, 26714657285316, 39117891024927, 3364138628143722
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For numerators see A145627. For general properties of A_l(x) see A145609.

Crossrefs

Cf. A145609 - A145640.

Programs

  • Maple
    A := proc(l,x) add(x^(l-d)/d,d=1..l-1) ; end: A145629 := proc(n) denom( A(2*n+1,10)) ; end: seq(A145629(n),n=1..20) ; # R. J. Mathar, Aug 21 2009
  • Mathematica
    m = 10; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Denominator[k]], {r, 1, 30}]; aa (* Artur Jasinski *)

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145614 Denominator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=3.

Original entry on oeis.org

2, 4, 20, 280, 280, 3080, 5720, 2288, 116688, 3695120, 25865840, 594914320, 2974571600, 8923714800, 86262576400, 5348279736800, 5348279736800, 16044839210400, 197886350261600, 15222026943200, 89157586381600
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For numerators see A145613. For general properties of A_l(x) see A145609.

Crossrefs

Cf. A145609 - A145640.

Programs

  • Mathematica
    m = 3; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Denominator[k]], {r, 1, 30}]; aa (*Artur Jasinski*)

Extensions

Edited by R. J. Mathar, Aug 21 2009

A145615 Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=4.

Original entry on oeis.org

18, 883, 23566, 5278979, 380087174, 66895348819, 13914232622662, 178102177617521, 4036982692723202, 6136213692944321089, 32726473029037904778, 72260052448115886127009, 2890402097924635887833902
Offset: 1

Views

Author

Artur Jasinski, Oct 14 2008

Keywords

Comments

For denominators see A145616. For general properties of A_l(x) see A145609.

Crossrefs

Cf. A145609 - A145640.

Programs

  • Mathematica
    m = 4; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (* Artur Jasinski *)
a[n_,m_]:=Integrate[(m-x^n)/(m-x),{x,0,1}]+(m^n-m)Log[m/(m-1)]Table[4 a[2 n, 4] // FullSimplify  // Numerator, {n,1,25}]  (* Gerry Martens , Jun 04 2016 *)

Extensions

Edited by R. J. Mathar, Aug 21 2009
Previous Showing 11-20 of 40 results. Next

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