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A289307 Coefficients in expansion of E_4^(1/4) in powers of q. 14
1, 60, -4860, 660480, -105063420, 18206269560, -3328461434880, 631226199152640, -122944850563477500, 24436796345920143420, -4935178772322020730360, 1009598430837232126725120, -208736157503462405753487360, 43541664791244563211024015480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.

R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

FORMULA

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/4).

G.f.: 2F1(1/12, 5/12; 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 06 2017 [See also the Kontsevich and Zagier link, where t = 1728/j = 1 - Sum_{k>=0} A289210(k)*q^k, with q = q(z) = exp(2*Pi*I*z), Im(z) > 0. - Wolfdieter Lang, May 27 2018]

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = sqrt(3) * Gamma(1/3)^(9/2) * Gamma(1/4) / (16 * 2^(3/4) * Pi^4) = 0.201967785736579402060958871696381229013432952780653381728912717635... - Vaclav Kotesovec, Jul 07 2017, updated Mar 04 2018

EXAMPLE

From Seiichi Manyama, Jul 07 2017: (Start)

2F1(1/12, 5/12; 1; 1728/j)

= 1 + (1*5)/(1*1) * 12/j + (1*5*13*17)/(1*1*2*2) * (12/j)^2 + (1*5*13*17*25*29)/(1*1*2*2*3*3) * (12/j)^3 + ...

= 1 + 60/j + 39780/j^2 + 38454000/j^3 + ...

= 1 + 60*q - 44640*q^2 + 21399120*q^3 - ...

           + 39780*q^2 - 59192640*q^3 + ...

                       + 38454000*q^3 - ...

                                      + ...

= 1 + 60*q -  4860*q^2 +   660480*q^3 - ... (End)

MATHEMATICA

a[ n_] := SeriesCoefficient[ ComposeSeries[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, q], {q, 0, n}], q^2 / Series[q^2 KleinInvariantJ[ Log[q]/(2 Pi I)], {q, 0, n}]], {q, 0, n}]; (* Michael Somos, Jun 21 2018 *)

CROSSREFS

E_4^(k/8): A108091 (k=1), this sequence (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A000521 (j), A004009 (E_4), A066395 (1/j), A092870, A110163, A289210.

Sequence in context: A293091 A074076 A084274 * A091032 A328951 A178785

Adjacent sequences:  A289304 A289305 A289306 * A289308 A289309 A289310

KEYWORD

sign

AUTHOR

Seiichi Manyama, Jul 02 2017

STATUS

approved

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Last modified April 9 12:51 EDT 2020. Contains 333352 sequences. (Running on oeis4.)