A289308 -id:A289308 - cp's OEIS frontend

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

1, 30, -2880, 416640, -69178110, 12378401280, -2321610157440, 449733567736320, -89200812128140800, 18013245273252679710, -3689479088922151082880, 764375901202388789804160, -159862757100127037505991680, 33699694000689939789618455040, -7152050326608893289997995966720, 1526705794390267864554876727856640

Offset: 0

N. J. A. Sloane and Michael Somos, Jun 06 2005

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			More precisely, the theta series of E_8 begins 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + ... and the 8th root of this is 1 + 30*q^2 - 2880*q^4 + 416640*q^6 - 69178110*q^8 + ...

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Seiichi Manyama, Table of n, a(n) for n = 0..424
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, Seven Staggering Sequences.

E_4^(k/8): this sequence (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5).

Cf. A001158, A004009 (E_4), A106205, A110163, A300147, A303007.

nmax = 20; s = 8; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/16), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

R. = PowerSeriesRing(ZZ,20)
a = R(eisenstein_series_qexp(4,20, normalization='integral'))
list(a.sqrt().sqrt().sqrt()) # Andy Huchala, Jul 10 2021

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/8). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 3^(1/4) * Gamma(1/3)^(9/4) / (2^(33/8) * Pi^(3/2) * Gamma(7/8)) = 0.1141392450598624077174159151600898926678394937157356242319309115... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303007(k) * (-f(q))^k where f(q) is Sum_{k>=1} sigma_3(k)*q^k. - Seiichi Manyama, Jun 15 2018

A289292 Coefficients in expansion of E_4^(1/2).

Original entry on oeis.org

1, 120, -6120, 737760, -107249640, 17385063120, -3014720249760, 547287510713280, -102701836021530600, 19762301660609250840, -3878226140959368843120, 773209219953012480001440, -156173318001506652330786720, 31888935085481430265623676560

Offset: 0

Seiichi Manyama, Jul 02 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..425
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29b).

E_k^(1/2): A289291 (k=2), this sequence (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).
E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), this sequence (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).
Cf. A001421, A004009 (E_4), A110163.

terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/2).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 3*Gamma(1/3)^9 / (32*sqrt(2)*Pi^(13/2)) = 0.27646925986847687648926173728588572192308632719... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
G.f.: 3F2(1/6, 1/2, 5/6; 1, 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

Original entry on oeis.org

1, 60, -4860, 660480, -105063420, 18206269560, -3328461434880, 631226199152640, -122944850563477500, 24436796345920143420, -4935178772322020730360, 1009598430837232126725120, -208736157503462405753487360, 43541664791244563211024015480

Offset: 0

Seiichi Manyama, Jul 02 2017

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			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)

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).

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.

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 *)

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

A289309 Coefficients in expansion of E_4^(5/8).

Original entry on oeis.org

1, 150, -5400, 625200, -86672550, 13570016400, -2289741037200, 406440122001600, -74830416797043000, 14162747887897808550, -2738995393669565720400, 538973037306449327998800, -107578899914865970323788400, 21729813219122500082762389200

Offset: 0

Seiichi Manyama, Jul 02 2017

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

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), this sequence (k=5), A289318 (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(5/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(13/8), where c = 5 * 3^(5/4) * Gamma(1/3)^(45/4) / (256 * 2^(5/8) * Pi^(15/2) * Gamma(3/8)) = 0.2571085249207580781634342667473393997795373224370302803101380883544... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289319 Coefficients in expansion of E_4^(7/8).

Original entry on oeis.org

1, 210, -1260, 232680, -28907970, 4211355960, -671557897080, 113817372354240, -20151698294479500, 3687092782592216970, -692109989731133096760, 132609267059636375116920, -25838624519733523814390760, 5105657091664960508653858680

Offset: 0

Seiichi Manyama, Jul 02 2017

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In general, for 0 < m < 1, the expansion of (E_4)^m is asymptotic to (-1)^(n+1) * m * 3^(2*m) * Gamma(1/3)^(18*m) * exp(Pi*sqrt(3)*n) / (2^(9*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - Vaclav Kotesovec, Mar 05 2018

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

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), this sequence (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(7/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(7*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(15/8), where c = 7 * 3^(7/4) * Gamma(1/3)^(63/4) / (1024 * 2^(7/8) * Pi^(21/2) * Gamma(1/8)) = 0.1121182787986009012644546699220584282491804117887058146553161217384... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289318 Coefficients in expansion of E_4^(3/4).

Original entry on oeis.org

1, 180, -3780, 447840, -59046660, 8921092680, -1463828444640, 253953515257920, -45858209756343300, 8534765953624978260, -1626301691950399586280, 315807346469727624396960, -62284193156782292089690080, 12443904711281870749228431240

Offset: 0

Seiichi Manyama, Jul 02 2017

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

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), this sequence (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(3/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(3*A110163(n)/4).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(7/4), where c = 3^(5/2) * Gamma(1/3)^(27/2) / (256 * 2^(3/4) * Pi^9 * Gamma(1/4)) = 0.2007048471908800363193160136812560289856774734680572658944418664975... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289247 Coefficients in expansion of 1/E_4^(1/8).

Original entry on oeis.org

1, -30, 3780, -616440, 111056910, -21135698280, 4165203862440, -840914061328320, 172810940671692900, -35998781800053352710, 7579904611028433074280, -1609957152292592382408360, 344417407415742189796786680, -74127324674775434904036905640

Offset: 0

Seiichi Manyama, Jul 08 2017

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

E_4^(k/8): A001943 (k=-8), A289566 (k=-4), A295815 (k=-2), this sequence (k=-1), A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8).

Cf. A110163, A300147.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(-1/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-A110163(n)/8).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(7/8), where c = Pi^(3/2) / (2^(15/8) * 3^(1/4) * Gamma(1/3)^(9/4) * Gamma(9/8)) = 0.133402757019143151407904538533... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018

A299955 Coefficients in expansion of E_4^(3/2).

Original entry on oeis.org

1, 360, 24840, -465120, 57417480, -6800282640, 930889890720, -139401582644160, 22250341370421000, -3723955494287559480, 646515765251485521840, -115559140273640812421280, 21150946022800731753255840, -3948247836773858791840263120

Offset: 0

Seiichi Manyama, Feb 22 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..426

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8), this sequence (k=12), A008410 (k=16), A008411 (k=24), A282012 (k=32), A282015 (k=40).

Convolution cube of A289292.

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 81*Gamma(1/3)^27 / (32768*sqrt(2)*Pi^(37/2)) = 0.39832876770813443250501819621900549862424768734... - Vaclav Kotesovec, Mar 05 2018

cp's OEIS Frontend

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

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A289292 Coefficients in expansion of E_4^(1/2).

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A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

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A289309 Coefficients in expansion of E_4^(5/8).

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A289319 Coefficients in expansion of E_4^(7/8).

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A289318 Coefficients in expansion of E_4^(3/4).

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A289247 Coefficients in expansion of 1/E_4^(1/8).

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A299955 Coefficients in expansion of E_4^(3/2).

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