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A106205
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Expansion of (q*j(q))^(1/24) where j(q) is the elliptic modular invariant (A000521).
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22
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1, 31, -2848, 413823, -68767135, 12310047967, -2309368876639, 447436508910495, -88755684988520798, 17924937024841839390, -3671642907594608226078, 760722183234128461061246, -159105706560247952472114973
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OFFSET
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0,2
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COMMENTS
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For k > 0, if mod(k,8) <> 0 then (q*j(q))^(k/24) is asymptotic to -(-1)^n * sin(k*Pi/8) * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * Gamma(k/8) * exp(Pi*sqrt(3)*n) / (Pi^(k/2 + 1) * 2^(k/8 + 3) * exp(k*Pi/(8*sqrt(3))) * n^(k/8 + 1)). Equivalently, is asymptotic to -(-1)^n * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * exp(Pi*sqrt(3)*(n - k/24)) / (Pi^(k/2) * 2^(k/8 + 3) * Gamma(1 - k/8) * n^(k/8 + 1)).
For k > 0, if mod(k,8) = 0 then (q*j(q))^(k/24) is asymptotic to exp(Pi*sqrt(2*k*n/3)) * k^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).
(End)
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LINKS
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FORMULA
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This is essentially the eighth root of the theta series of E_8 (A108091), divided by the Dedekind eta function. - N. J. A. Sloane, Aug 08 2005
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 0.11364889078525240958152388212499254894082832445224690827436413842337... = 3^(1/8) * sqrt(2 - sqrt(2)) * Gamma(1/8) * Gamma(1/3)^(3/4) / (2^(33/8) * exp(Pi/(8 * sqrt(3))) * Pi^(3/2)). - Vaclav Kotesovec, Jul 02 2017, updated Mar 06 2018
a(n) * A289397(n) ~ c * exp(2*Pi*sqrt(3)*n) / n^2, where c = -sqrt(2-sqrt(2)) / (16*Pi). - Vaclav Kotesovec, Mar 06 2018
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EXAMPLE
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1 + 31*q - 2848*q^2 + 413823*q^3 - 68767135*q^4 + 12310047967*q^5 - 2309368876639*q^6 + ...
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MATHEMATICA
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CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(1/8) / (2*QPochhammer[-1, x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)
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PROG
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(PARI) {a(n)=if(n<0, 0, polcoeff( (ellj(x+x^2*O(x^n))*x)^(1/24), n))}
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CROSSREFS
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(q*j(q))^(k/24): this sequence (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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