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.

Previous Showing 11-17 of 17 results.

A289294 Coefficients in expansion of E_10^(1/2).

Original entry on oeis.org

1, -132, -76428, -12686784, -4629945804, -1581036186312, -643032851554368, -264454897726360704, -114830224962140965068, -50847479367845783084484, -23070238839261012248537688, -10629338992044523324726971456
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

E_k^(1/2): A289291 (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), this sequence (k=10), A289295 (k=14).
Cf. A013974 (E_10), A289024.

Programs

  • Mathematica
    nmax = 20; s = 10; CoefficientList[Series[Sqrt[1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A289024(n)/2).
a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3^(3/2) * Pi^(5/2) / (2^(9/2) * Gamma(3/4)^12) = -0.3503612261281732359954402284478780636268623476628... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A289566 Coefficients in expansion of 1/E_4^(1/2).

Original entry on oeis.org

1, -120, 20520, -3934560, 793510440, -164694615120, 34824089129760, -7460017581785280, 1613575314347164200, -351613291994820018840, 77073167391611232305520, -16975579813113940564868640, 3753822590560913900129106720
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), this sequence (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
Cf. A001943 (1/E_4), A110163, A289292 (E_4^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,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)^(-A110163(n)/2).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / sqrt(n), where c = 3^(7/2) * Gamma(2/3)^9 / (2^(9/2) * Pi^(7/2)) = 0.5756695813762774104492155417156662666189119445257965... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

A110150 G.f.: 4th root of Eisenstein series E_10 (cf. A013974).

Original entry on oeis.org

1, -66, -40392, -9009264, -3725341158, -1400292801072, -604993149612720, -262280205541007808, -118717180239835505592, -54520207050101542651506, -25525844887805197307977968, -12095360676632550886664063760, -5797006133905562955666277287792, -2803076705590018145443840156918512
Offset: 0

Views

Author

N. J. A. Sloane, Sep 15 2005

Keywords

Links

Crossrefs

E_k^(1/4): A289392 (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), this sequence (k=10), A289391 (k=14).
Cf. A004984, A013974, A109817, A289294.

Programs

  • Mathematica
    nmax = 20; s = 10; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

Formula

a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3^(3/4) * Pi^(3/2) / (2^(15/4) * Gamma(3/4)^7) = -0.227361380713650977567497769428903183591275821407342369621... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
G.f.: Sum_{k>=0} A004984(k) * (33*f(q))^k where f(q) is Sum_{k>=1} sigma_9(k)*q^k. - Seiichi Manyama, Jun 16 2018

A289295 Coefficients in expansion of E_14^(1/2).

Original entry on oeis.org

1, -12, -98388, -20312544, -5889254484, -2083830070392, -810894400450848, -334381509272710464, -143464412162723380308, -63364234685240118242604, -28614423885137875351570248, -13150804531745894256074689056
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

E_k^(1/2): A289291 (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), this sequence (k=14).
Cf. A058550 (E_14), A289029.

Programs

  • Mathematica
    nmax = 20; s = 14; CoefficientList[Series[Sqrt[1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(A289029(n)/2).
a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -9 * Pi^(7/2) / (2^(11/2) * Gamma(3/4)^16) = -0.422728335899452596724927626919867458580193404969719... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A289247 Coefficients in expansion of 1/E_4^(1/8).

Original entry on oeis.org

1, -30, 3780, -616440, 111056910, -21135698280, 4165203862440, -840914061328320, 172810940671692900, -35998781800053352710, 7579904611028433074280, -1609957152292592382408360, 344417407415742189796786680, -74127324674775434904036905640
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

E_4^(k/8): A001943 (k=-8), A289566 (k=-4), A295815 (k=-2), this sequence (k=-1), A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8).
Cf. A110163, A300147.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(-1/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A110163(n)/8).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(7/8), where c = Pi^(3/2) / (2^(15/8) * 3^(1/4) * Gamma(1/3)^(9/4) * Gamma(9/8)) = 0.133402757019143151407904538533... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018

A289391 Coefficients in expansion of E_14^(1/4).

Original entry on oeis.org

1, -6, -49212, -10451544, -4218246978, -1581565900392, -677142351901080, -293172823731286848, -132241381826055031692, -60651805300034501958126, -28350123351848675673466968, -13420046900399367136336144200
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2017

Keywords

Links

Crossrefs

E_k^(1/4): A289392 (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), this sequence (k=14).
Cf. A004984, A058550 (E_14).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[13, 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)^(A289029(n)/4).
a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3*Pi^2 / (2^(17/4) * Gamma(3/4)^9) = -0.2497407198517688195944362279691013167903920989625478927175764401875... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_13(k)*q^k. - Seiichi Manyama, Jun 16 2018

A299955 Coefficients in expansion of E_4^(3/2).

Original entry on oeis.org

1, 360, 24840, -465120, 57417480, -6800282640, 930889890720, -139401582644160, 22250341370421000, -3723955494287559480, 646515765251485521840, -115559140273640812421280, 21150946022800731753255840, -3948247836773858791840263120
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

Keywords

Links

Crossrefs

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8), this sequence (k=12), A008410 (k=16), A008411 (k=24), A282012 (k=32), A282015 (k=40).

Formula

Convolution cube of A289292.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 81*Gamma(1/3)^27 / (32768*sqrt(2)*Pi^(37/2)) = 0.39832876770813443250501819621900549862424768734... - Vaclav Kotesovec, Mar 05 2018
Previous Showing 11-17 of 17 results.

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

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