A289307 -id:A289307 - cp's OEIS frontend

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

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

Offset: 0

Seiichi Manyama, Jul 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..367

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).

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 *)

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

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

Seiichi Manyama, Feb 13 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..424

Cf. A004009 (E_4), A289247, A289307.

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

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

Seiichi Manyama, Feb 22 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..426

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).

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

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

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

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

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