A289340 -id:A289340 - 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

A289339 Coefficients of (q*(j(q)-1728))^(7/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -287, -42595, -9750370, -3081185660, -1117168154431, -438204467218406, -181018051263504195, -77584080248087108885, -34183723168674046275385, -15388633770558568711781905, -7047808475666778827478858184

Offset: 0

Seiichi Manyama, Jul 02 2017

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(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), this sequence (k=7), A289340 (k=8), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(7/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(7*A289061(k)/24).

a(n) ~ c * exp(2*Pi*n) / n^(19/12), where c = -7 * exp(-7*Pi/12) * Gamma(1/12) / (2^(35/12) * 3^(1/12) * Pi^(17/12) * Gamma(3/4)^(1/3)) = -0.287342744567300675294730727139553541489784437990631575713791583301655... - Vaclav Kotesovec, Mar 07 2018

cp's OEIS Frontend

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

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