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 1-4 of 4 results.

A294974 Coefficients in expansion of (E_2^4/E_4)^(1/8).

Original entry on oeis.org

1, -42, 4032, -659904, 118064226, -22406634432, 4407587356032, -888750999070464, 182478248639753472, -37986867560948245674, 7994272624037726124672, -1697243410477799687716416, 362963150140702802158191360, -78095916585903527021840348352
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2018

Keywords

Comments

Also coefficients in expansion of (E_2^8/E_8)^(1/16).

Crossrefs

Programs

  • Mathematica
    terms = 14;
    E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
    E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
    (E2[x]^4/E4[x])^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^A294626(n).
a(n) ~ (-1)^n * 2^(13/8) * Pi * exp(Pi*sqrt(3)*n) / (Gamma(1/8) * Gamma(1/3)^(9/4) * n^(7/8)). - Vaclav Kotesovec, Jun 03 2018

A295815 Coefficients in expansion of E_4^(-1/4).

Original entry on oeis.org

1, -60, 8460, -1459680, 273388620, -53595097560, 10818138134880, -2228446076600640, 465957083177325900, -98553257565313635420, 21034800052217022675960, -4522762142866403196901920, 978397734079422399475947360
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2018

Keywords

Crossrefs

Cf. A004009 (E_4), A289247, A289307.

Formula

Convolution square of A289247.
Convolution inverse of A289307.
a(n) ~ (-1)^n * 2^(9/4) * Pi^3 * exp(Pi*sqrt(3)*n) / (sqrt(3) * Gamma(1/3)^(9/2) * Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 05 2018

A300147 a(n) = (1/8) * Sum_{d|n} d * A110163(d).

Original entry on oeis.org

-30, 6660, -1536120, 354476040, -81800478900, 18876653594640, -4356063194112240, 1005225129672310800, -231970363216834560390, 53530545369975222475800, -12352954264801690636800360, 2850624405442199478575792160
Offset: 1

Views

Author

Seiichi Manyama, Feb 26 2018

Keywords

Crossrefs

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / 8. - Vaclav Kotesovec, Jun 07 2018

A378469 Coefficients in expansion of (1/E_4)^4.

Original entry on oeis.org

1, -960, 567360, -266138880, 108735481920, -40500351480960, 14114830665358080, -4678563821426250240, 1491145606587529742400, -460511820740945555286720, 138585483759128030100927360, -40812342463218781348220286720, 11800049457060387849887324117760, -3358272262154871467174772417214080
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 27 2024

Keywords

Comments

In general, for k > 0, the expansion of 1/(E_4)^k is asymptotic to (-1)^n * k * 2^(9*k) * Pi^(12*k) * n^(k-1) * exp(Pi*sqrt(3)*n) / (3^(2*k) * Gamma(1/3)^(18*k) * Gamma(k+1)).

Crossrefs

Cf. A001943 (k=1), A287933 (k=2), A378468 (k=3).
Cf. A289566 (k=1/2), A295815 (k=1/4), A289247 (k=1/8).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1+240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])^(-4), {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * 34359738368 * Pi^48 * n^3 * exp(Pi*sqrt(3)*n) / (19683 * Gamma(1/3)^72).
Showing 1-4 of 4 results.