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A289325 Coefficients in expansion of E_6^(1/6). 14
1, -84, -20412, -6617856, -2505409788, -1027549673640, -442991672331264, -197605206331169280, -90359564898413083644, -42105781947560460595284, -19913609001700051596476280, -9531377528273693889501019392 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.30).

FORMULA

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/6).

G.f.: 2F1(1/12, 7/12; 1; 1728/(1728-j)) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017

a(n) ~ c * exp(2*Pi*n) / n^(7/6), where c = -Gamma(1/4)^(8/3) * Gamma(1/3)^2 / (2^(9/2) * 3^(1/6) * Pi^(7/2)) = -0.149083170913265334790743918765758886634155... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

EXAMPLE

From Seiichi Manyama, Jul 08 2017: (Start)

2F1(1/12, 7/12; 1; 1728/(1728 - j))

= 1 - A289557(1)/(j - 1728) + A289557(2)/(j - 1728)^2 - A289557(3)/(j - 1728)^3 + ...

= 1 - 84/(j - 1728) + 62244/(j - 1728)^2 - 64318800/(j - 1728)^3 + ...

= 1 - 84*q - 82656*q^2 -  64795248*q^3 - ...

           + 62244*q^2 + 122496192*q^3 + ...

                       -  64318800*q^3 - ...

                                       + ...

= 1 - 84*q - 20412*q^2 -   6617856*q^3 - ... (End)

MATHEMATICA

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])^(1/6), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

CROSSREFS

E_6^(k/12): A109817 (k=1), this sequence (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A013973 (E_6), A288851, A289417, A289557.

Sequence in context: A278725 A186245 A224429 * A202923 A232914 A145495

Adjacent sequences:  A289322 A289323 A289324 * A289326 A289327 A289328

KEYWORD

sign

AUTHOR

Seiichi Manyama, Jul 02 2017

STATUS

approved

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Last modified July 13 05:18 EDT 2020. Contains 335673 sequences. (Running on oeis4.)