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 25 results. Next

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

A289326 Coefficients in expansion of E_6^(1/4).

Original entry on oeis.org

1, -126, -27972, -8603784, -3156774138, -1265670056952, -536028623834760, -235629947944839168, -106414175763732002292, -49052892961209924090486, -22977990271885179647877768, -10904016663130642099838196120
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), this sequence (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/4).
a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -sqrt(3) * Gamma(1/4)^5 / (32 * 2^(3/4) * Pi^4) = -0.20698746071805886655919194203910626895689130674662074751291... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289325 Coefficients in expansion of E_6^(1/6).

Original entry on oeis.org

1, -84, -20412, -6617856, -2505409788, -1027549673640, -442991672331264, -197605206331169280, -90359564898413083644, -42105781947560460595284, -19913609001700051596476280, -9531377528273693889501019392
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Examples

			From _Seiichi Manyama_, Jul 08 2017: (Start)
2F1(1/12, 7/12; 1; 1728/(1728 - j))
= 1 - A289557(1)/(j - 1728) + A289557(2)/(j - 1728)^2 - A289557(3)/(j - 1728)^3 + ...
= 1 - 84/(j - 1728) + 62244/(j - 1728)^2 - 64318800/(j - 1728)^3 + ...
= 1 - 84*q - 82656*q^2 -  64795248*q^3 - ...
           + 62244*q^2 + 122496192*q^3 + ...
                       -  64318800*q^3 - ...
                                       + ...
= 1 - 84*q - 20412*q^2 -   6617856*q^3 - ... (End)

Links

Crossrefs

E_6^(k/12): A109817 (k=1), this sequence (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851, A289417, A289557.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/6), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/6).
G.f.: 2F1(1/12, 7/12; 1; 1728/(1728-j)) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017
a(n) ~ c * exp(2*Pi*n) / n^(7/6), where c = -Gamma(1/4)^(8/3) * Gamma(1/3)^2 / (2^(9/2) * 3^(1/6) * Pi^(7/2)) = -0.149083170913265334790743918765758886634155... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289327 Coefficients in expansion of E_6^(1/3).

Original entry on oeis.org

1, -168, -33768, -9806496, -3482370024, -1364023149552, -567278132268960, -245678241438057792, -109559333350138970088, -49951945835561166375048, -23173552482577051154061168, -10901813191731667585777068000
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), this sequence (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/3), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/3).
a(n) ~ c * exp(2*Pi*n) / n^(4/3), where c = -3^(1/6) * Gamma(1/4)^(16/3) * Gamma(1/3) / (32 * 2^(1/3) * Pi^5) = -0.25096087408563316781920388861983614789... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289392 Coefficients in expansion of E_2^(1/4).

Original entry on oeis.org

1, -6, -72, -1104, -20238, -405792, -8601840, -189317568, -4281478272, -98841343686, -2318973049008, -55118876238000, -1324194430710912, -32099173821105312, -784045854628721568, -19276683937074656064, -476644852188898489662
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2017

Keywords

Links

Crossrefs

E_2^(k/4): this sequence (k=1), A289291 (k=2), A289393 (k=3).
E_k^(1/4): this sequence (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), A289391 (k=14).
Cf. A004984, A006352 (E_2), A108091, A109817, A289394.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^A289394(n).
a(n) ~ c / (n^(5/4) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.209452682241344640265132676904094736935029272937832600102950644347... - Vaclav Kotesovec, Jul 08 2017
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_1(k)*q^k. - Seiichi Manyama, Jun 16 2018

A289328 Coefficients in expansion of E_6^(5/12).

Original entry on oeis.org

1, -210, -37800, -10300080, -3534651750, -1351633962672, -551776752641520, -235367241169341120, -103623939263346377400, -46723958347194591810690, -21464711387762586693907248, -10009787904868201520473221840
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), this sequence (k=5), A289293 (k=6).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(5/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(5*A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(17/12), where c = -5 * Gamma(1/12) * Gamma(1/4)^(20/3) / (128 * 2^(11/12) * 3^(2/3) * Pi^5 * Gamma(1/3)^2) = -0.2792181117471536554156263079143941137076647484619917046386429000631... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289345 Coefficients in expansion of E_6^(7/12).

Original entry on oeis.org

1, -294, -40572, -9456216, -3013531458, -1095736644072, -430427492908056, -177966281438573376, -76323096421188881292, -33643171872410204427918, -15150435131179232328586968, -6940567145625149028384495432
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), this sequence (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(7/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(7*A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(19/12), where c = -7 * Gamma(1/12) * Gamma(1/4)^(22/3) / (1024 * 6^(1/12) * Pi^7) = -0.2836006135316422535659652380776952016594933981... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289346 Coefficients in expansion of E_6^(2/3).

Original entry on oeis.org

1, -336, -39312, -8266944, -2529479568, -895678457184, -344891780549568, -140330667583849344, -59379605532142099344, -25873741825665005773200, -11534062764689844375098592, -5236325710480558290644292672
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), this sequence (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(2/3), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(2*A288851(n)/3).
a(n) ~ c * exp(2*Pi*n) / n^(5/3), where c = -3^(1/3) * Gamma(1/4)^(32/3) / (128 * 2^(2/3) * Pi^8 * Gamma(1/3)) = -0.258650618394676269905172499217587002338... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289347 Coefficients in expansion of E_6^(3/4).

Original entry on oeis.org

1, -378, -36288, -6664896, -1950813774, -672039262944, -253536117254784, -101485291597998336, -42360328701954544176, -18242860786892766495450, -8049299329628263783504512, -3621056234759774113947852096
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), this sequence (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(3/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(3*A288851(n)/4).
a(n) ~ c * exp(2*Pi*n) / n^(7/4), where c = -3^(5/2) * Gamma(1/4)^11 / (2048 * 2^(3/4) * Pi^9) = -0.21604472104032272720247495618663130188448925463945370445... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018
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