A303055 -id:A303055 - cp's OEIS frontend

A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

1, -42, -11088, -3774624, -1472710974, -617481728640, -270883381218912, -122585272771463040, -56747118995519331456, -26727350506044696990762, -12760853360973370821796320, -6159994719956314185540737376, -3000691311646502407278581263104, -1472883416501251994527873967792256

Offset: 0

N. J. A. Sloane, Sep 15 2005

sign

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

E_6^(k/12): this sequence (k=1), A289325 (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. A001160, A013973, A288851, A299503, A303055.

nmax = 20; s = 6; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/12). - Seiichi Manyama, Jul 02 2017
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(10/3) * Gamma(1/3)^2 / (16 * 6^(1/12) * Pi^3 * Gamma(1/12)) = -0.079329971529325538458906713053582098... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
Equivalently, c = -Gamma(1/3) * Gamma(1/4)^(7/3) / (2^(23/6) * 3^(11/24) * sqrt(1 + sqrt(3)) * Pi^(5/2)). - Vaclav Kotesovec, Aug 03 2025
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303055(k) * f(q)^k where f(q) is Sum_{k>=1} sigma_5(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

sign

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

A303007 Expansion of (1-240*x)^(1/8).

Original entry on oeis.org

1, -30, -3150, -472500, -81506250, -15160162500, -2956231687500, -595469525625000, -122815589660156250, -25791273828632812500, -5493541325498789062500, -1183608449221102734375000, -257434837705589844726562500, -56437637496994696728515625000

Offset: 0

Seiichi Manyama, Jun 15 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A003557, A007947, A049210, A108091.

(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), A004996 (b=36), this sequence (b=240), A303055 (b=504), A305886 (b=1728).

CoefficientList[Series[Surd[1-240x,8],{x,0,20}],x] (* Harvey P. Dale, Aug 29 2024 *)

N=20; x='x+O('x^N); Vec((1-240*x)^(1/8))

a(n) = 30^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = 15^n * A301271(n).
a(n) ~ -2^(4*n - 3) * 15^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +30*(-8*n+9)*a(n-1)=0. - 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

sign

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

A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A303007 Expansion of (1-240*x)^(1/8).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula