A303007 -id:A303007 - 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

A301271 Expansion of (1-16*x)^(1/8).

Original entry on oeis.org

1, -2, -14, -140, -1610, -19964, -259532, -3485144, -47920730, -670890220, -9526641124, -136837208872, -1984139528644, -28998962341720, -426699017313880, -6315145456245424, -93937788661650682, -1403541077650545484, -21053116164758182260, -316904801216886322440

Offset: 0

Seiichi Manyama, Jun 15 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..833

Cf. A003557, A007947.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), this sequence (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

PARI
```
N=20; x='x+O('x^N); Vec((1-16*x)^(1/8))
```

a(n) = 2^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = -sqrt(2-sqrt(2)) * Gamma(1/8) * Gamma(n-1/8) * 16^(n-1) / (Pi*Gamma(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) ~ -2^(4*n-3) / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +2*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = -2*A097184(n-1). - R. J. Mathar, Jan 20 2020

A305991 Expansion of (1-27*x)^(1/9).

Original entry on oeis.org

1, -3, -36, -612, -11934, -250614, -5513508, -125235396, -2911722957, -68910776649, -1653858639576, -40143659706072, -983519662798764, -24285370135261788, -603664914790793016, -15091622869769825400, -379177024602966863175, -9568643738510163782475

Offset: 0

Seiichi Manyama, Jun 16 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..700

Cf. A003557, A007947.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), this sequence (b=27), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

PARI
```
N=20; x='x+O('x^N); Vec((1-27*x)^(1/9))
```

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k - 1) for n > 0.
a(n) ~ 27^n / (Gamma(-1/9) * n^(10/9)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +3*(-9*n+10)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

cp's OEIS Frontend

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

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A301271 Expansion of (1-16*x)^(1/8).

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A305991 Expansion of (1-27*x)^(1/9).

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