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 11 results. Next

A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

Original entry on oeis.org

1, -42, -11088, -3774624, -1472710974, -617481728640, -270883381218912, -122585272771463040, -56747118995519331456, -26727350506044696990762, -12760853360973370821796320, -6159994719956314185540737376, -3000691311646502407278581263104, -1472883416501251994527873967792256
Offset: 0

Views

Author

N. J. A. Sloane, Sep 15 2005

Keywords

Links

Crossrefs

E_6^(k/12): this sequence (k=1), A289325 (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. A001160, A013973, A288851, A299503, A303055.

Programs

  • Mathematica
    nmax = 20; s = 6; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/12). - Seiichi Manyama, Jul 02 2017
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(10/3) * Gamma(1/3)^2 / (16 * 6^(1/12) * Pi^3 * Gamma(1/12)) = -0.079329971529325538458906713053582098... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
Equivalently, c = -Gamma(1/3) * Gamma(1/4)^(7/3) / (2^(23/6) * 3^(11/24) * sqrt(1 + sqrt(3)) * Pi^(5/2)). - Vaclav Kotesovec, Aug 03 2025
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303055(k) * f(q)^k where f(q) is Sum_{k>=1} sigma_5(k)*q^k. - Seiichi Manyama, Jun 15 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

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

A289349 Coefficients in expansion of E_6^(11/12).

Original entry on oeis.org

1, -462, -24948, -2518824, -654112074, -212483064024, -76819071738024, -29728723632736128, -12066341379893331300, -5073593348593538950566, -2192302482140061697816872, -968086916154014421082349304, -435126775136273350146250044888
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Comments

In general, for 0 < m < 1, the expansion of (E_6)^m is asymptotic to -m * Gamma(1/4)^(16*m) * 3^(2*m) * exp(2*Pi*n) / (2^(13*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - Vaclav Kotesovec, Mar 05 2018

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), A289347 (k=9), A289348 (k=10), this sequence (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(11/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(11*A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(23/12), where c = -11 * 2^(5/12) * 3^(5/6) * Pi^(11/3) / (128 * Gamma(1/12) * Gamma(3/4)^(44/3)) = -0.08406022472181281739983743854923746657261382508944840919197295490535... - Vaclav Kotesovec, Jul 08 2017

A289567 Coefficients in expansion of 1/E_6^(1/2).

Original entry on oeis.org

1, 252, 103572, 46355904, 21754545876, 10493652271032, 5153897870227008, 2563741466120209536, 1287429765611338091988, 651251466581383330576956, 331360676706818772917367912, 169399388595923901462013678656
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), this sequence (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
E_6^(k/12): A289570 (k=-18), A000706 (k=-12), this sequence (k=-6), A109817 (k=1), A289325 (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. A000706 (1/E_6), A288851, A289293 (E_6^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A288851(n)/2).
a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 2^(5/2) * Gamma(3/4)^8 / (3*Pi^(5/2)) = 0.5480868931611627439175185425300450785609564636925943866686455998197... - Vaclav Kotesovec, Jul 09 2017, updated Mar 03 2018

A289570 Coefficients in expansion of 1/E_6^(3/2).

Original entry on oeis.org

1, 756, 501228, 311671584, 187266950892, 110121960638088, 63808586297102304, 36578013578688141504, 20797655630223547290348, 11749541312124028845092052, 6603568491137827506152966712, 3695593478842608407829235523808
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

E_6^(k/12): this sequence (k=-18), A000706 (k=-12), A289567 (k=-6), A109817 (k=1), A289325 (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. A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-3/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-3*A288851(n)/2).
a(n) ~ c * exp(2*Pi*n) * sqrt(n), where c = 2^(17/2) * Gamma(3/4)^24 / (27 * Pi^(13/2)) = 1.0344943380746471723299237298670710161068814236907171661035... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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