A299829 -id:A299829 - cp's OEIS frontend

A289300 Expansion of (q*j(q))^(5/24) where j(q) is the elliptic modular invariant (A000521).

1, 155, -4630, 601265, -83644610, 13148835656, -2223584717035, 395257299676190, -72843145114522035, 13796578308407774725, -2669652272250261922223, 525556527400692937755655, -104937908072571416700653120

Offset: 0

Seiichi Manyama, Jul 02 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), this sequence (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(5/8) / (2*QPochhammer[-1, x])^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(5/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A192731(n)/24).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(13/8), where c = 0.251632947646443757912747944865268710111059274679945447776728146817... = 5 * 3^(5/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(15/4) * Gamma(5/8) / (2^(37/8) * exp(5 * Pi / (8 * sqrt(3))) * Pi^(7/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299829(n) ~ -5*sqrt(2 + sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Feb 20 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

A289300 Expansion of (q*j(q))^(5/24) where j(q) is the elliptic modular invariant (A000521).

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