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 1-6 of 6 results.

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

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

A289210 Coefficients in expansion of E_6^2/E_4^3.

Original entry on oeis.org

1, -1728, 1285632, -616294656, 242544070656, -85253786824320, 27846073156184064, -8638345400999827968, 2579332695698905989120, -747814048389765750131136, 211795259563761765262894080, -58852853362216364363212075776
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 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), 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), this sequence (k=288).
Cf. A000521 (j), A004009 (E_4), A013973 (E_6), A066395, A289209, A300025.
E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

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, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

a(n) = -1728 * A066395(n) for n > 0.
G.f.: 1 - 1728 * q * Product_{k>=1} (1-q^k)^24 / E_4^3 = 1 - 1728/j.
G.f.: (E_6*E_6)/(E_4*E_8) = (E_6*E_10)/(E_8*E_8). - Seiichi Manyama, Jun 29 2017
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n^2, where c = 256 * Pi^12 / Gamma(1/3)^18 = 4.684993039417145659090436569582265840407909701042523126716193567422... - 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

A300147 a(n) = (1/8) * Sum_{d|n} d * A110163(d).

Original entry on oeis.org

-30, 6660, -1536120, 354476040, -81800478900, 18876653594640, -4356063194112240, 1005225129672310800, -231970363216834560390, 53530545369975222475800, -12352954264801690636800360, 2850624405442199478575792160
Offset: 1

Views

Author

Seiichi Manyama, Feb 26 2018

Keywords

Links

Crossrefs

Cf. A004009 (E_4), A108091, A110163, A289247, A300025.

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / 8. - Vaclav Kotesovec, Jun 07 2018

A299503 a(n) = (1/12) * Sum_{d|n} d * A288851(d).

Original entry on oeis.org

42, 23940, 12795048, 6852216840, 3669291602172, 1964875343509008, 1052174343447263568, 563430581238674063376, 301712374716950167413282, 161564459029576395778765080, 86516419639708839110100858360, 46328820782943003562067180265504
Offset: 1

Views

Author

Seiichi Manyama, Feb 26 2018

Keywords

Crossrefs

Cf. A013973 (E_6), A109817, A288851, A289540, A300025.
Showing 1-6 of 6 results.

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