A299831 -id:A299831 - cp's OEIS frontend

A299414 Coefficients in expansion of (E_6^2/E_4^3)^(1/3).

1, -576, 96768, -30253824, 4526272512, -1917275819904, 105679295281152, -161582272076127744, -20815321809392861184, -20529723592970845750080, -6560883968194298456036352, -3617226648349298247150473472

Offset: 0

Seiichi Manyama, Feb 21 2018

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

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), this sequence (k=96), A299413 (k=144), A289210 (k=288).

Cf. A000521 (j), A289340, A299831.

terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms; (E6[x]^2/E4[x]^3)^(1/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 22 2018 *)

G.f.: (1 - 1728/j)^(1/3), where j is the j-function.
a(n) ~ -Gamma(1/4)^(8/3) * exp(2*Pi*n) / (2^(5/3) * 3^(2/3) * Pi^2 * Gamma(1/3) * n^(5/3)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A300054(n) ~ -exp(4*Pi*n) / (sqrt(3)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299826 Coefficients in expansion of (q*j(q))^(-1/12) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, -62, 8579, -1476538, 276299401, -54140398258, 10925052030358, -2250028212438240, 470403050272649518, -99482921702360817662, 21231436164082720565341, -4564732260005808181200000, 987422026920066412423809840

Offset: 0

Seiichi Manyama, Feb 20 2018

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(q*j(q))^(k/24): A289397 (k=-1), this sequence (k=-2), A299827 (k=-3), A299828 (k=-4), A299829 (k=-5), A299830 (k=-6), A299831 (k=-8), A299832 (k=-12).

Cf. A000521, A289297.

CoefficientList[Series[(2 * QPochhammer[-1, x])^2 / (65536 + x*QPochhammer[-1, x]^24)^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 20 2018 *)

Convolution inverse of A289297.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(3/4), where c = 0.28101701912289268934379724324854717406285519051128823261445... = 2^(1/4) * exp(Pi/(4 * sqrt(3))) * Pi / (3^(1/4) * Gamma(1/4) * Gamma(1/3)^(3/2)). - Vaclav Kotesovec, Feb 20 2018, updated Mar 06 2018
a(n) * A289297(n) ~ -exp(2*sqrt(3)*n*Pi) / (2^(5/2)*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

cp's OEIS Frontend

A299414 Coefficients in expansion of (E_6^2/E_4^3)^(1/3).

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