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.

Previous Showing 11-19 of 19 results.

A299951 Coefficients in expansion of (E_4^3/E_6^2)^(1/16).

Original entry on oeis.org

1, 108, 18792, 8775216, 3375768096, 1535055129576, 691959629136096, 325485731190285792, 154751723387164258560, 74822912718767823810204, 36526619326785857845042608, 17998154668247683887778684176, 8931078840823632559970453020032
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

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), this sequence (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299857.

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}];
(E4[x]^3/E6[x]^2)^(1/16) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299857.
a(n) ~ sqrt(2) * Pi^(3/8) * exp(2*Pi*n) / (3^(1/16) * Gamma(1/8) * sqrt(Gamma(1/4)) * n^(7/8)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299857(n) ~ -sin(Pi/8) * exp(4*Pi*n) / (8*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299953 Coefficients in expansion of (E_4^3/E_6^2)^(1/12).

Original entry on oeis.org

1, 144, 27648, 12540096, 4971036672, 2263040955360, 1031452724072448, 487587831652591488, 233267529030162186240, 113311495859272029716688, 55566291037565862262794240, 27487705978359515260636550208, 13689979692617556597746930024448
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

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), this sequence (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299858.

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}];
(E4[x]^3/E6[x]^2)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299858.
a(n) ~ 2^(2/3) * sqrt(Pi) * exp(2*Pi*n) / (3^(1/12) * Gamma(1/6) * Gamma(1/4)^(2/3) * n^(5/6)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299858(n) ~ -exp(4*Pi*n) / (12*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299993 Coefficients in expansion of (E_4^3/E_6^2)^(1/9).

Original entry on oeis.org

1, 192, 41472, 18342144, 7524397056, 3440911653504, 1589472997005312, 756816895536990720, 364982499184388898816, 178417371665487543380928, 88017286719942539086814208, 43770603489875525093472688896, 21905830503405563891572154843136
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

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), this sequence (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299863.

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}];
(E4[x]^3/E6[x]^2)^(1/9) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299863.
a(n) ~ 2^(8/9) * Pi^(2/3) * exp(2*Pi*n) / (3^(1/9) * Gamma(2/9) * Gamma(1/4)^(8/9) * n^(7/9)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299863(n) ~ -2*sin(2*Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299994 Coefficients in expansion of (E_4^3/E_6^2)^(1/8).

Original entry on oeis.org

1, 216, 49248, 21609504, 9000122112, 4129083886032, 1919370450227328, 917374442680570176, 444151666318727522304, 217813424092164713883960, 107771776495186976967396672, 53736084111333058216805911392, 26958647064591216695092188902400
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

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), this sequence (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299859.

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}];
(E4[x]^3/E6[x]^2)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299859.
a(n) ~ 2 * Pi^(3/4) * exp(2*Pi*n) / (3^(1/8) * Gamma(1/4)^2 * n^(3/4)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299859(n) ~ -exp(4*Pi*n) / (4*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300052 Coefficients in expansion of (E_4^3/E_6^2)^(1/6).

Original entry on oeis.org

1, 288, 76032, 33042816, 14318032896, 6651157620672, 3146793694792704, 1522045714678435584, 745464270665241870336, 369134048335617435664800, 184269983601798163049283072, 92610644166133510115124717696
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2018

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), this sequence (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299860.

Programs

  • Mathematica
    terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/6) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299860.
a(n) ~ 2^(4/3) * Pi * exp(2*Pi*n) / (3^(1/6) * Gamma(1/4)^(4/3) * Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299860(n) ~ -exp(4*Pi*n) / (2*sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300053 Coefficients in expansion of (E_4^3/E_6^2)^(1/4).

Original entry on oeis.org

1, 432, 145152, 64494144, 29760915456, 14274670230432, 6975951829890048, 3459591515857458816, 1733116511275051696128, 875135886353582630388336, 444632598699435462934282752, 227042568315636603738176892096
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2018

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), this sequence (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299861.

Programs

  • Mathematica
    terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/4) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299861.
a(n) ~ 4 * Pi * exp(2*Pi*n) / (3^(1/4) * Gamma(1/4)^2 * sqrt(n)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299861(n) ~ -exp(4*Pi*n) / (2*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300054 Coefficients in expansion of (E_4^3/E_6^2)^(1/3).

Original entry on oeis.org

1, 576, 235008, 109880064, 53449592832, 26574124961664, 13393739222599680, 6814262482916285952, 3490692930294883909632, 1797524713443792341369664, 929454499859725260939506688, 482202319224911188610453541120
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2018

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), this sequence (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299414.

Programs

  • Mathematica
    terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299414.
a(n) ~ 2^(8/3) * Pi^2 * exp(2*Pi*n) / (3^(1/3) * Gamma(1/4)^(8/3) * Gamma(2/3) * n^(1/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299414(n) ~ -exp(4*Pi*n) / (sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A300055 Coefficients in expansion of (E_4^3/E_6^2)^(1/2).

Original entry on oeis.org

1, 864, 476928, 254399616, 136313874432, 72985679394624, 39084426149704704, 20929208813297429760, 11207444175842517172224, 6001488285356611750823136, 3213747681163891383409648128, 1720934927015053152217599326592
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2018

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), this sequence (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299413.

Programs

  • Mathematica
    terms = 12;
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}];
(E4[x]^3/E6[x]^2)^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299413.
a(n) ~ 16 * Pi^3 * exp(2*Pi*n) / (sqrt(3) * Gamma(1/4)^4). - Vaclav Kotesovec, Mar 04 2018

A289417 Coefficients of 1/(q*(j(q)-1728)) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 984, 771372, 543802432, 361216628430, 230920762687776, 143732944930479800, 87718753215371355648, 52729710063184125105381, 31319171802847165756090320, 18421996714811488321383528228, 10748837396953435386200311855872
Offset: 0

Views

Author

Seiichi Manyama, Jul 06 2017

Keywords

Links

Crossrefs

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), A289561 (k=-48), this sequence (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).
Cf. A289061, A289209.

Programs

  • Mathematica
    CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A289061(n)).
a(n) ~ c * exp(2*Pi*n) * n, where c = Gamma(3/4)^8 * exp(2*Pi) / (324 * Pi^2) = 0.851487576721136974981670736748581778120097667011853803210435262759745... - Vaclav Kotesovec, Mar 07 2018
Previous Showing 11-19 of 19 results.

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