A145640 -id:A145640 - cp's OEIS frontend

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.

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

Offset: 1

Artur Jasinski, Oct 14 2008

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

Cf. A145609 - A145640.

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

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 *)

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

Artur Jasinski, Oct 14 2008

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

Cf. A145609 - A145640.

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

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 *)

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

Artur Jasinski, Oct 14 2008

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

Robert Israel, Table of n, a(n) for n = 1..390

Cf. A145609 - A145640.

f:= n -> numer(add(7^(2*n+1-d)/d, d=1..2*n)):
map(f, [$1..40]); # Robert Israel, Jun 05 2016

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 *)

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

Artur Jasinski, Oct 14 2008

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

Robert Israel, Table of n, a(n) for n = 1..375

Cf. A145609 - A145640.

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

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 *)

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

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

Artur Jasinski, Oct 14 2008

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

Robert Israel, Table of n, a(n) for n = 1..1150

Cf. A145609 - A145640.

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

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 *)

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

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

Artur Jasinski, Oct 14 2008

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

Cf. A145609 - A145640.

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

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 *)

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

Artur Jasinski, Oct 14 2008

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

Cf. A145609 - A145640.

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*)

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

Artur Jasinski, Oct 14 2008

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

Cf. A145609 - A145640.

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 *)

Edited by R. J. Mathar, Aug 21 2009

A145616 Denominator 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

1, 3, 5, 70, 315, 3465, 45045, 36036, 51051, 4849845, 1616615, 223092870, 557732175, 5019589575, 145568097675, 36100888223400, 410237366175, 410237366175, 15178782548475, 30357565096950, 622330084487475

Offset: 1

Artur Jasinski, Oct 14 2008

For numerators see A145615. For general properties of A_l(x) see A145609.

Cf. A145609 - A145640.

m = 4; 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*)

Edited by R. J. Mathar, Aug 21 2009

A145617 Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=5.

Original entry on oeis.org

55, 8365, 209195, 73218955, 5491423277, 1510141416085, 490795960391965, 24539798019883535, 10429414158454786655, 4953971725266096561953, 11259026648332043641555, 6473940322790925219990095

Offset: 1

Artur Jasinski, Oct 14 2008

For denominators see A145618. For general properties of A_l(x) see A145609.

Cf. A145609 - A145640.

m = 5; 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[5 a[2 n, 5] // FullSimplify  // Numerator, {n,1,25}]  (* Gerry Martens , Jun 04 2016 *)

Edited by R. J. Mathar, Aug 21 2009

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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