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.

A299856 Coefficients in expansion of (E_6^2/E_4^3)^(1/18).

Original entry on oeis.org

1, -96, -6912, -5410944, -1537640448, -753077521728, -307254291047424, -143134552425743616, -65142005576276164608, -30798673631132393592288, -14628259811568672073824768, -7054762801208507859522653568
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), this sequence (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 = 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}];
(E6[x]^2/E4[x]^3)^(1/18) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/18), where j is the j-function.
a(n) ~ c * exp(2*Pi*n) / n^(10/9), where c = -Gamma(1/4)^(4/9) / (2^(4/9) * 3^(35/18) * Pi^(1/3) * Gamma(8/9)) = -0.0974650059642735838539936939997471425... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299950(n) ~ -sin(Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299857 Coefficients in expansion of (E_6^2/E_4^3)^(1/16).

Original entry on oeis.org

1, -108, -7128, -5975856, -1648702944, -817564231656, -330392410226208, -154125342449733600, -69899495093389741824, -33019122368612611954332, -15654348707682435222420432, -7540807164973158284078993424
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), this sequence (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 = 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}];
(E6[x]^2/E4[x]^3)^(1/16) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/16), where j is the j-function.
a(n) ~ -3^(1/16) * sqrt(Gamma(1/4)) * exp(2*Pi*n) / (8 * sqrt(2) * Pi^(3/8) * Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299951(n) ~ -sin(Pi/8) * exp(4*Pi*n) / (8*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299858 Coefficients in expansion of (E_6^2/E_4^3)^(1/12).

Original entry on oeis.org

1, -144, -6912, -7563456, -1885022208, -979976901600, -383134788854784, -179914112738674560, -80649007527361757184, -38019764211792500474064, -17921855069499640580651520, -8604055343353988623666807872
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), this sequence (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 = 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}];
(E6[x]^2/E4[x]^3)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/12), where j is the j-function.
a(n) ~ -Gamma(1/4)^(2/3) * exp(2*Pi*n) / (2^(5/3) * 3^(11/12) * sqrt(Pi) * Gamma(5/6) * n^(7/6)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299953(n) ~ -exp(4*Pi*n) / (12*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299859 Coefficients in expansion of (E_6^2/E_4^3)^(1/8).

Original entry on oeis.org

1, -216, -2592, -10412064, -1955812608, -1193816824272, -424976182312320, -205525905843878208, -89308328381644142592, -42098146869799454214456, -19580168925118916335723968, -9345687920591466548039096160
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), this sequence (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j).

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

Formula

G.f.: (1 - 1728/j)^(1/8), where j is the j-function.
a(n) ~ -3^(1/8) * Gamma(1/4) * exp(2*Pi*n) / (8 * Pi^(3/4) * Gamma(3/4) * n^(5/4)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299994(n) ~ -exp(4*Pi*n) / (4*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299860 Coefficients in expansion of (E_6^2/E_4^3)^(1/6).

Original entry on oeis.org

1, -288, 6912, -13136256, -1543993344, -1312510191552, -400771816381440, -207423640540991232, -85808962187667652608, -40835278880374417949088, -18652791316021929491056128, -8871850083830324974981015680
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), this sequence (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j).

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

Formula

G.f.: (1 - 1728/j)^(1/6), where j is the j-function.
a(n) ~ -Gamma(1/4)^(4/3) * exp(2*Pi*n) / (2^(4/3) * 3^(5/6) * Pi * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A300052(n) ~ -exp(4*Pi*n) / (2*sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299861 Coefficients in expansion of (E_6^2/E_4^3)^(1/4).

Original entry on oeis.org

1, -432, 41472, -19704384, 593104896, -1488746462112, -215673487239168, -180545262418802304, -58940991594820435968, -31030127172303490499184, -13143520096697989968012288, -6336110261914309914844683456
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), this sequence (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
Cf. A000521 (j), A289334, A299830.

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

Formula

G.f.: (1 - 1728/j)^(1/4), where j is the j-function.
a(n) ~ -3^(1/4) * Gamma(1/4)^2 * exp(2*Pi*n) / (8 * Pi^2 * n^(3/2)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A300053(n) ~ -exp(4*Pi*n) / (2*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299862 Coefficients in expansion of (E_6^2/E_4^3)^(1/32).

Original entry on oeis.org

1, -54, -5022, -3259116, -1012953978, -479848911192, -201506019745716, -93655132040105136, -43096009052844972522, -20449878102745826555178, -9772372681245342509703768, -4732826670479844302345499132, -2309711500786845517082643561660
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), this sequence (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)^(1/32) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/32), where j is the j-function.
a(n) ~ c * exp(2*Pi*n) / n^(17/16), where c = -3^(1/32) * Gamma(1/4)^(1/4) / (2^(17/4) * Pi^(3/16) * Gamma(15/16)) = -0.0582176906417343821471376177620947... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299949(n) ~ -sin(Pi/16) * exp(4*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299863 Coefficients in expansion of (E_6^2/E_4^3)^(1/9).

Original entry on oeis.org

1, -192, -4608, -9494784, -1988603904, -1136127187584, -419383041398784, -200225564597488128, -88040635024586342400, -41470393697874515307456, -19381646100387803980004352, -9267227811160245194038205184
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), this sequence (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 = 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}];
(E6[x]^2/E4[x]^3)^(1/9) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/9), where j is the j-function.
a(n) ~ -2^(1/9) * Gamma(1/4)^(8/9) * exp(2*Pi*n) / (3^(17/9) * Pi^(2/3) * Gamma(7/9) * n^(11/9)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299993(n) ~ -2*sin(2*Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

Original entry on oeis.org

1, 60, -4860, 660480, -105063420, 18206269560, -3328461434880, 631226199152640, -122944850563477500, 24436796345920143420, -4935178772322020730360, 1009598430837232126725120, -208736157503462405753487360, 43541664791244563211024015480
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Examples

			From _Seiichi Manyama_, Jul 07 2017: (Start)
2F1(1/12, 5/12; 1; 1728/j)
= 1 + (1*5)/(1*1) * 12/j + (1*5*13*17)/(1*1*2*2) * (12/j)^2 + (1*5*13*17*25*29)/(1*1*2*2*3*3) * (12/j)^3 + ...
= 1 + 60/j + 39780/j^2 + 38454000/j^3 + ...
= 1 + 60*q - 44640*q^2 + 21399120*q^3 - ...
           + 39780*q^2 - 59192640*q^3 + ...
                       + 38454000*q^3 - ...
                                      + ...
= 1 + 60*q -  4860*q^2 +   660480*q^3 - ... (End)

Links

Crossrefs

E_4^(k/8): A108091 (k=1), this sequence (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).
Cf. A000521 (j), A004009 (E_4), A066395 (1/j), A092870, A110163, A289210.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ ComposeSeries[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, q], {q, 0, n}], q^2 / Series[q^2 KleinInvariantJ[ Log[q]/(2 Pi I)], {q, 0, n}]], {q, 0, n}]; (* Michael Somos, Jun 21 2018 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/4).
G.f.: 2F1(1/12, 5/12; 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 06 2017 [See also the Kontsevich and Zagier link, where t = 1728/j = 1 - Sum_{k>=0} A289210(k)*q^k, with q = q(z) = exp(2*Pi*I*z), Im(z) > 0. - Wolfdieter Lang, May 27 2018]
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = sqrt(3) * Gamma(1/3)^(9/2) * Gamma(1/4) / (16 * 2^(3/4) * Pi^4) = 0.201967785736579402060958871696381229013432952780653381728912717635... - Vaclav Kotesovec, Jul 07 2017, updated Mar 04 2018
Previous Showing 11-19 of 19 results.

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