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.

A289348 Coefficients in expansion of E_6^(5/6).

Original entry on oeis.org

1, -420, -31500, -4724160, -1314429900, -440028142344, -162555920654400, -63990327056960640, -26341675849615282380, -11210298679649742846180, -4895195936831699458605912, -2181913188022929464292248640
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), A289347 (k=9), this sequence (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}])^(5/6), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(5*A288851(n)/6).
a(n) ~ c * exp(2*Pi*n) / n^(11/6), where c = -5 * 3^(1/6) * Gamma(1/4)^(40/3) / (2048*sqrt(2) * Pi^(19/2) * Gamma(1/3)^2) = -0.1571123439957640423587958439875289712533650298096956968521099309872... - 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

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

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

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

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
Previous Showing 11-17 of 17 results.

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