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 1-10 of 19 results. Next

A289366 Coefficients in expansion of (E_6^2/E_4^3)^(1/288).

Original entry on oeis.org

1, -6, -702, -393804, -132734778, -61428055320, -26480146877172, -12318952616296752, -5730786812846192490, -2732960583228848850522, -1314627022075990658598360, -639871947654492158944455132, -313833506047227501170833823292
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2017

Keywords

Comments

In general, for 0 < m < 1/2, the expansion of (E_6^2/E_4^3)^m is asymptotic to -m * 3^m * Gamma(1/4)^(8*m) * exp(2*n*Pi) / (2^(8*m-1) * Pi^(6*m) * Gamma(1-2*m) * n^(1+2*m)). - Vaclav Kotesovec, Mar 04 2018

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): this sequence (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A066395 (1/j), A289365, A289367, A300025, A305886.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[((1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3)^(1/288), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: (1 - 1728/j)^(1/288).
G.f.: Product_{n>=1} (1-q^n)^A289367(n).
a(n) ~ c * exp(2*Pi*n) / n^(145/144), where c = -Gamma(1/4)^(1/36) / (48 * 2^(1/36) * 3^(287/288) * Pi^(1/48) * Gamma(143/144)) = -0.006892157290355982837398273285864980110980721215574657372422958228077... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 25 2018
a(n) * A289365(n) ~ -sin(Pi/144) * exp(4*Pi*n) / (144*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A289209 Coefficients in expansion of E_4^3/E_6^2.

Original entry on oeis.org

1, 1728, 1700352, 1332930816, 939690602496, 624182333927040, 399031077924476928, 248370528839869094400, 151578005556161702559744, 91116938989182168182098368, 54119528875319902426524072960, 31833210323194251819350736777984
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2017

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), this sequence (k=288).
Cf. A004009 (E_4), A013973 (E_6), A289063, A289210, A289417, A300025.
E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: 1 + 1728 * q * Product_{k>=1} (1-q^k)^24 / E_6^2.
G.f.: (E_4*E_8)/(E_6*E_6) = (E_8*E_8)/(E_6*E_10). - Seiichi Manyama, Jun 29 2017
a(n) = 1728 * A289417(n - 1) for n > 0. - Seiichi Manyama, Jul 08 2017
a(n) ~ c * exp(2*Pi*n) * n, where c = 256 * Pi^6 / (3 * Gamma(1/4)^8) = 2.747700206704861755142526128354171788550012833617513654955480535522... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(0) = 1, a(n) = (288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 2018

A299413 Coefficients in expansion of (E_6^2/E_4^3)^(1/2).

Original entry on oeis.org

1, -864, 269568, -75240576, 19930724352, -5124295980864, 1292387210099712, -321604751662509312, 79241739168490536960, -19376923061550541800672, 4709786462808256974509568, -1139188440993923671697455488
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), this sequence (k=144), A289210 (k=288).
Cf. A000521 (j), A007242, A299832.

Programs

  • Mathematica
    terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms; (E6[x]^2/E4[x]^3)^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 22 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/2), where j is the j-function.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * sqrt(n), where c = 32*sqrt(2) * Pi^(11/2) / Gamma(1/3)^9. - Vaclav Kotesovec, Mar 04 2018

A289368 Coefficients in expansion of (E_6^2/E_4^3)^(1/24).

Original entry on oeis.org

1, -72, -6048, -4217184, -1264437504, -606533479920, -251777443450752, -117085712395216320, -53634689421870422016, -25408429618361083967592, -12110787335129301116994240, -5854620911089647830793873696
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2017

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), this sequence (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).
Cf. A289366, A289367, A294974, A294976.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[((1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3)^(1/24), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: (1 - 1728/j)^(1/24).
G.f.: Product_{n>=1} (1-q^n)^(12*A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(1/3) / (2^(7/3) * 3^(23/24) * Pi^(1/4) * Gamma(11/12)) = -0.07569217204117312767729284017524325060022536591050774997610261275428... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(n) * A289369(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A296609 Coefficients in expansion of (E_6^2/E_4^3)^(1/144).

Original entry on oeis.org

1, -12, -1368, -779184, -260251104, -120710392488, -51881715871776, -24129355507367136, -11210568318996090624, -5342692661136883228860, -2567906908021088206807248, -1249094126109188331384940944, -612254304549600491293149962880
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), this sequence (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A299694.

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]^2/E4[x]^3)^(1/144) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/144).
a(n) ~ -Gamma(1/4)^(1/18) * exp(2*Pi*n) / (24 * 2^(1/18) * 3^(143/144) * Pi^(1/24) * Gamma(71/72) * n^(73/72)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299694(n) ~ -sin(Pi/72) * exp(4*Pi*n) / (72*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A296614 Coefficients in expansion of (E_6^2/E_4^3)^(1/96).

Original entry on oeis.org

1, -18, -1998, -1156356, -382624794, -177898412808, -76229340502932, -35444571049682064, -16446161396159063082, -7832755937588033655054, -3761678744155185551186328, -1828621496185972561746774324, -895757692814150533920101726460
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), this sequence (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A299696.

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]^2/E4[x]^3)^(1/96) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/96).
a(n) ~ -Gamma(1/4)^(1/12) * exp(2*Pi*n) / (16 * 2^(1/12) * 3^(95/96) * Pi^(1/16) * Gamma(47/48) * n^(49/48)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299696(n) ~ -sin(Pi/48) * exp(4*Pi*n) / (48*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A296652 Coefficients in expansion of (E_6^2/E_4^3)^(1/72).

Original entry on oeis.org

1, -24, -2592, -1525536, -499930368, -233042911056, -99547207597440, -46277719207526208, -21444241881136232448, -10206632934331485363576, -4897739115250118143468992, -2379385980983995218900931680, -1164826509542958652906666171392
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), this sequence (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j).

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]^2/E4[x]^3)^(1/72) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/72).
a(n) ~ -Gamma(1/4)^(1/9) * exp(2*Pi*n) / (12 * 2^(1/9) * 3^(71/72) * Pi^(1/12) * Gamma(35/36) * n^(37/36)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299697(n) ~ -sin(Pi/36) * exp(4*Pi*n) / (36*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A297021 Coefficients in expansion of (E_6^2/E_4^3)^(1/48).

Original entry on oeis.org

1, -36, -3672, -2240784, -719628768, -337401534456, -143188210269216, -66549102831096480, -30753876262814297856, -14619380361359418716724, -7003704012123711964880592, -3398241529278572532519050928, -1661531038403129009358413705856
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), this sequence (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j).

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]^2/E4[x]^3)^(1/48) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/48).
a(n) ~ -Gamma(1/4)^(1/6) * exp(2*Pi*n) / (8 * 2^(1/6) * 3^(47/48) * Pi^(1/8) * Gamma(23/24) * n^(25/24)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299698(n) ~ -sin(Pi/24) * exp(4*Pi*n) / (24*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299414 Coefficients in expansion of (E_6^2/E_4^3)^(1/3).

Original entry on oeis.org

1, -576, 96768, -30253824, 4526272512, -1917275819904, 105679295281152, -161582272076127744, -20815321809392861184, -20529723592970845750080, -6560883968194298456036352, -3617226648349298247150473472
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), this sequence (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A289340, A299831.

Programs

  • Mathematica
    terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms; (E6[x]^2/E4[x]^3)^(1/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 22 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/3), where j is the j-function.
a(n) ~ -Gamma(1/4)^(8/3) * exp(2*Pi*n) / (2^(5/3) * 3^(2/3) * Pi^2 * Gamma(1/3) * n^(5/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A300054(n) ~ -exp(4*Pi*n) / (sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299422 Coefficients in expansion of (E_6^2/E_4^3)^(1/36).

Original entry on oeis.org

1, -48, -4608, -2926656, -919916544, -434180785824, -182989456349184, -85043754451706496, -39190139442556010496, -18607302407649844554480, -8899353903793993480829952, -4312672556860403013966227136, -2105991149652021429396842987520
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), this sequence (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j).

Programs

  • Mathematica
    terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]^2/E4[x]^3)^(1/36) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/36), where j is the j-function.
a(n) ~ c * exp(2*Pi*n) / n^(19/18), where c = -Gamma(1/4)^(2/9) / (2^(11/9) * 3^(71/36) * Pi^(1/6) * Gamma(17/18)) = -0.0521763497905021090549912315961203... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299943(n) ~ -sin(Pi/18) * exp(4*Pi*n) / (18*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018
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