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 1-5 of 5 results.

A300025 a(n) = Sum_{d|n} d * A289367(d).

Original entry on oeis.org

6, 1440, 1194264, 541478400, 312591006756, 162166557492864, 88044200553447984, 46868779675750146048, 25162028756680583497806, 13459244040350535046357440, 7210731049497803400061471560, 3860497513211463446882383706112
Offset: 1

Views

Author

Seiichi Manyama, Feb 25 2018

Keywords

Links

Crossrefs

Cf. A289365, A289366, A289367.

Formula

a(n) ~ exp(2*Pi*n) / 144. - Vaclav Kotesovec, Jun 03 2018

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

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

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
Showing 1-5 of 5 results.

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