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 1-7 of 7 results.

A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)).

Original entry on oeis.org

1, 1, 5, 27, 169, 1115, 7760, 55748, 411498, 3101490, 23785645, 185064559, 1457664666, 11602828475, 93205739436, 754751603157, 6155229065861, 50515624923790, 416930705579538, 3458726257239312, 28825340825747729, 241245120218823892, 2026803168946440648
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Links

Crossrefs

Cf. A258268, A258788, A258791, A258793, A258794.

Programs

  • Maple
    T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n
  • Mathematica
    Table[SeriesCoefficient[1/Product[x^(2*k)*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(n+2)}], {n, 0, 30}]

Formula

a(n) ~ c * d^n / n^2, where d = A258268 = 9.15337019245412246194853029240135454007332720412184884968926320147613... = r^3/(r-1), where r is the root of the equation polylog(2, 1-r) + 3*log(r)^2/2 = 0, c = 0.8069142856822510276258439534144172057548... .

A258790 a(n) = [x^n] Product_{k=1..n} 1/(x^(3*k)*(1-x^k)).

Original entry on oeis.org

1, 1, 6, 48, 411, 3765, 36308, 363446, 3742085, 39405777, 422669224, 4603472960, 50790334667, 566603884871, 6381702580969, 72481863380510, 829331355150992, 9551576115706329, 110654552651370400, 1288710163262774157, 15080440970246785366, 177237948953055593475
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Links

Crossrefs

Cf. A258788, A258790, A258792, A258794, A258795, A258796.

Programs

  • Maple
    T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n
  • Mathematica
    Table[SeriesCoefficient[1/Product[x^(3*k)*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(3*n+5)/2}], {n, 0, 25}]

Formula

a(n) ~ c * d^n / n^2, where d = 12.8718984948677835397002665286811919572579479691341210018008114644121... = r^4/(r-1), where r is the root of the equation polylog(2, 1-r) + 2*log(r)^2 = 0, c = 0.44720199058408831652920046766862756... .

A258791 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^2).

Original entry on oeis.org

1, 3, 20, 158, 1307, 11352, 102538, 954904, 9112038, 88723163, 878714118, 8829998320, 89848944237, 924291213496, 9600148608620, 100565064076006, 1061498376477423, 11281275452880277, 120635822090127386, 1297256892395670322, 14021436433125959714
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Crossrefs

Cf. A258788, A258789, A258793, A258794.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/Product[x^k*(1-x^k)^2, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 25}]

Formula

a(n) ~ c * d^n / n^(5/2), where d = 12.0708016857156441729965623654557363850943928675996965027830903372727... = r^3/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + (3*log(r)^2)/4 = 0, c = 8.36819319541... .

A258792 a(n) = [x^n] Product_{k=1..n} 1/(x^k*(1-x^k)^3).

Original entry on oeis.org

1, 6, 69, 915, 12978, 194688, 3051617, 49526487, 826910754, 14135805042, 246508115583, 4372617452085, 78714369892152, 1435357362134796, 26472477913596486, 493178852479545556, 9270953614684288962, 175695092091980786166, 3354069936616380522256
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Crossrefs

Cf. A258788, A258790, A258794, A258795, A258796.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/Product[x^k*(1-x^k)^3, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^3, {x, 0, n*(n+3)/2}], {n, 0, 20}]

Formula

a(n) ~ c * d^n / n^3, where d = 22.0610202494679061193859054301626736218023392292898139172609021542610... = r^4/(r-1)^3, where r is the root of the equation polylog(2, 1-r) + (2*log(r)^2)/3 = 0, c = 20.953639522741... .

A258793 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)^2).

Original entry on oeis.org

1, 4, 55, 896, 16494, 326422, 6812064, 147937628, 3315019979, 76184664934, 1787702723767, 42688437971038, 1034621287862521, 25398832816003228, 630502487733706193, 15805630063826901440, 399669931534045129915, 10184690536676439639278, 261340023300544414822171
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Crossrefs

Cf. A258788, A258789, A258791, A258794.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/Product[x^(2*k)*(1-x^k)^2, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^2, {x, 0, n*(n+2)}], {n, 0, 20}]

Formula

a(n) ~ c * d^n / n^(5/2), where d = 29.1694173246928561008040480794933198469510496062455151175744673506960... = r^4/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + log(r)^2 = 0, c = 2.0036140319464... .

A258795 a(n) = [x^n] Product_{k=1..n} 1/(x^(3*k)*(1-x^k)^2).

Original entry on oeis.org

1, 5, 112, 3216, 104112, 3661517, 136580866, 5323418568, 214685704402, 8897404908604, 377068336570902, 16280261371485594, 714081427614467553, 31747177836376617322, 1428084942303149795972, 64902413675181889657064, 2976483322906106920966911
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Crossrefs

Cf. A258788, A258790, A258791, A258792, A258793, A258794, A258796.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/Product[x^(3*k)*(1-x^k)^2, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^2, {x, 0, n*(3*n+5)/2}], {n, 0, 20}]

Formula

a(n) ~ c * d^n / n^(5/2), where d = 53.0676066669703028123492951828168330443393201750491213178019371417684... = r^5/(r-1)^2, where r is the root of the equation polylog(2, 1-r) + (5*log(r)^2)/4 = 0, c = 0.983501005499107... .

A258796 a(n) = [x^n] Product_{k=1..n} 1/(x^(3*k)*(1-x^k)^3).

Original entry on oeis.org

1, 15, 882, 67385, 5938518, 575782833, 59765085601, 6529604684991, 742474127495175, 87176531917206953, 10508492822243329854, 1294860745291809207237, 162553748258042032103013, 20735748733960087597815855, 2682101373558320853655174803
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Crossrefs

Cf. A258788, A258790, A258792, A258794, A258795.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/Product[x^(3*k)*(1-x^k)^3, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}]^3, {x, 0, n*(3*n+5)/2}], {n, 0, 20}]

Formula

a(n) ~ c * d^n / n^3, where d = 157.540286488430979726276374519534734829527107090287337321136938826336... = r^6/(r-1)^3, where r is the root of the equation polylog(2, 1-r) + log(r)^2 = 0, c = 1.797864597437050667... .
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.