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 11-16 of 16 results.

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

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

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

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

A289540 Coefficients in expansion of 1/E_6^(1/12).

Original entry on oeis.org

1, 42, 12852, 4780104, 1974512526, 863778376440, 391960077239304, 182430901827757632, 86505196617272556900, 41607881477457256661154, 20239469012268054187498440, 9935363620927698868439915544, 4914082482014906612773260362232
Offset: 0

Views

Author

Seiichi Manyama, Jul 15 2017

Keywords

Links

Crossrefs

E_6^(k/12): A289570 (k=-18), A000706 (k=-12), A289567 (k=-6), this sequence (k=-1), 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. A013973, A288851, A299503.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 26 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(5/12) * Gamma(3/4)^(4/3) / (3^(1/6) * Pi^(1/3) * Gamma(1/12)) = 0.08654217651555778130817946575840803466... - Vaclav Kotesovec, Jul 26 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018

A295816 Coefficients in expansion of E_6^(-1/4).

Original entry on oeis.org

1, 126, 43848, 17653104, 7691658282, 3503623887792, 1642412452485168, 785518220230462656, 381292445408455002744, 187197126720396903747630, 92737188888575341959000432, 46279260465924099748796201232, 23235412441447288670723222759280
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2018

Keywords

Links

Crossrefs

Cf. A013973 (E_6), A289326.

Formula

Convolution inverse of A289326.
a(n) ~ 2^(3/4) * sqrt(3) * Gamma(3/4)^5 * exp(2*Pi*n) / (3 * Pi^2 * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.