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 18 results. Next

A289365 Coefficients in expansion of (E_4^3/E_6^2)^(1/288).

Original entry on oeis.org

1, 6, 738, 402444, 138030342, 63625789080, 27583809566796, 12841110779519280, 5988752245273028886, 2859827345620916000346, 1377856546809576262931880, 671500179383482897207038108, 329754232921005442388958831684
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2017

Keywords

Comments

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

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): this sequence (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), A289209 (k=288).
Cf. A289209 (E_4^3/E_6^2), A289366, A289367, A300025.

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)^(1/288), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(143/144), where c = 2^(1/36) * Pi^(1/48) / (3^(1/288) * Gamma(1/144) * Gamma(1/4)^(1/36)) = 0.00699657322237604876174085217217686... - 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) * A289366(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

A289369 Coefficients in expansion of (E_4^3/E_6^2)^(1/24).

Original entry on oeis.org

1, 72, 11232, 5461344, 2029222656, 924074630640, 411487620614784, 192705317913673152, 91031590937141544960, 43814578627107100088424, 21291642032558036150652480, 10450287314646252538819378464, 5166676457072455262194208351232
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 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), this sequence (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), A289209 (k=288).
Cf. A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).
Cf. A289367, A294978, A294979.

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)^(1/24), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-12*A289367(n)).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(1/3) * Pi^(1/4) / (3^(1/24) * Gamma(1/12) * Gamma(1/4)^(1/3)) = 0.0907014320494145997187363667820553893... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(n) * A289368(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299694 Coefficients in expansion of (E_4^3/E_6^2)^(1/144).

Original entry on oeis.org

1, 12, 1512, 813744, 281434656, 129501949608, 56296822560480, 26218237904433888, 12242575532254540032, 5850239653863742634172, 2820869122426120317439152, 1375631026432164061822527120, 675950202173640832786529615232
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A296609.

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)^(1/144) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A296609.
a(n) ~ 2^(1/18) * Pi^(1/24) * exp(2*Pi*n) / (3^(1/144) * Gamma(1/72) * Gamma(1/4)^(1/18) * n^(71/72)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296609(n) ~ -sin(Pi/72) * exp(4*Pi*n) / (72*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299696 Coefficients in expansion of (E_4^3/E_6^2)^(1/96).

Original entry on oeis.org

1, 18, 2322, 1234116, 430292646, 197681749128, 86165040337452, 40145493017336976, 18768723217958523222, 8975036477140737601806, 4331009172188712335053032, 2113419430011730408087143924, 1039122180212218474089489166980
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A296614.

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)^(1/96) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A296614.
a(n) ~ 2^(1/12) * Pi^(1/16) * exp(2*Pi*n) / (3^(1/96) * Gamma(1/48) * Gamma(1/4)^(1/12) * n^(47/48)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296614(n) ~ -sin(Pi/48) * exp(4*Pi*n) / (48*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299697 Coefficients in expansion of (E_4^3/E_6^2)^(1/72).

Original entry on oeis.org

1, 24, 3168, 1663776, 584685312, 268219092816, 117214929608832, 54637244971358016, 25574598700199847936, 12238100148358426410360, 5910293921259795914011968, 2885917219371433467109558368, 1419817980186833008095972357120
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A296652.

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)^(1/72) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A296652.
a(n) ~ 2^(1/9) * Pi^(1/12) * exp(2*Pi*n) / (3^(1/72) * Gamma(1/36) * Gamma(1/4)^(1/9) * n^(35/36)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296652(n) ~ -sin(Pi/36) * exp(4*Pi*n) / (36*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299698 Coefficients in expansion of (E_4^3/E_6^2)^(1/48).

Original entry on oeis.org

1, 36, 4968, 2551824, 910405152, 416585268216, 182967944992992, 85373023607994528, 40055910812083687680, 19194979975339075406388, 9284600439037161721276848, 4539375955473797523355108272, 2236041702620444573315950439808
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A297021.

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)^(1/48) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A297021.
a(n) ~ 2^(1/6) * Pi^(1/8) * exp(2*Pi*n) / (3^(1/48) * Gamma(1/24) * Gamma(1/4)^(1/6) * n^(23/24)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A297021(n) ~ -sin(Pi/24) * exp(4*Pi*n) / (24*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

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

A299943 Coefficients in expansion of (E_4^3/E_6^2)^(1/36).

Original entry on oeis.org

1, 48, 6912, 3479616, 1259268096, 575044765344, 253777092387840, 118545813515338368, 55748828845833043968, 26753648919849657887472, 12960874757914028815661568, 6344939709971525751086888640, 3129285552537639403735326646272
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), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299422.

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)^(1/36) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299422.
a(n) ~ c * exp(2*Pi*n) / n^(17/18), where c = 2^(2/9) * Pi^(1/6) / (3^(1/36) * Gamma(1/4)^(2/9) * Gamma(1/18)) = 0.0588537525900341685779220592527938... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299422(n) ~ -sin(Pi/18) * exp(4*Pi*n) / (18*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299949 Coefficients in expansion of (E_4^3/E_6^2)^(1/32).

Original entry on oeis.org

1, 54, 7938, 3958956, 1442594502, 658201268952, 291148964582796, 136084851675471024, 64069809910723011222, 30769281599576554087722, 14917804015099613922436392, 7307669924831130556163175612, 3606311646826590340455185471940
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), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299862.

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)^(1/32) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299862.
a(n) ~ c * exp(2*Pi*n) / n^(15/16), where c = 2^(1/4) * Pi^(3/16) / (3^(1/32) * Gamma(1/4)^(1/4) * Gamma(1/16)) = 0.06666699751397812787469360011212565... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299862(n) ~ -sin(Pi/16) * exp(4*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018
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