A004984 -id:A004984 - cp's OEIS frontend

A289392 Coefficients in expansion of E_2^(1/4).

1, -6, -72, -1104, -20238, -405792, -8601840, -189317568, -4281478272, -98841343686, -2318973049008, -55118876238000, -1324194430710912, -32099173821105312, -784045854628721568, -19276683937074656064, -476644852188898489662

Offset: 0

Seiichi Manyama, Jul 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..702

E_2^(k/4): this sequence (k=1), A289291 (k=2), A289393 (k=3).
E_k^(1/4): this sequence (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), A289391 (k=14).
Cf. A004984, A006352 (E_2), A108091, A109817, A289394.

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^A289394(n).
a(n) ~ c / (n^(5/4) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.209452682241344640265132676904094736935029272937832600102950644347... - Vaclav Kotesovec, Jul 08 2017
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_1(k)*q^k. - Seiichi Manyama, Jun 16 2018

A110150 G.f.: 4th root of Eisenstein series E_10 (cf. A013974).

Original entry on oeis.org

1, -66, -40392, -9009264, -3725341158, -1400292801072, -604993149612720, -262280205541007808, -118717180239835505592, -54520207050101542651506, -25525844887805197307977968, -12095360676632550886664063760, -5797006133905562955666277287792, -2803076705590018145443840156918512

Offset: 0

N. J. A. Sloane, Sep 15 2005

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Vaclav Kotesovec, Table of n, a(n) for n = 0..360
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_k^(1/4): A289392 (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), this sequence (k=10), A289391 (k=14).

Cf. A004984, A013974, A109817, A289294.

nmax = 20; s = 10; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3^(3/4) * Pi^(3/2) / (2^(15/4) * Gamma(3/4)^7) = -0.227361380713650977567497769428903183591275821407342369621... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

G.f.: Sum_{k>=0} A004984(k) * (33*f(q))^k where f(q) is Sum_{k>=1} sigma_9(k)*q^k. - Seiichi Manyama, Jun 16 2018

A048779 Coefficients of power series for (1 - (1-8*x)^(1/4))/2.

Original entry on oeis.org

1, 3, 14, 77, 462, 2926, 19228, 129789, 894102, 6258714, 44379972, 318056466, 2299792908, 16755634044, 122874649656, 906200541213, 6716545187814, 50000947509282, 373691291911476, 2802684689336070, 21086865757861860, 159109987082048580, 1203701641403324040

Offset: 1

Jan Kristian Haugland

			G.f.: x + 3*x^2 + 14*x^3 + 77*x^4 + 462*x^5 + 2926*x^6 + 19228*x^7 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..1112
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Related to Catalan numbers (A000108).

Cf. A004984, A068465.

[Round(8^(n-1)*Gamma(n-1/4)/(Gamma(3/4)*Gamma(n+1))): n in [1..40]]; // G. C. Greubel, Aug 09 2022

a[ n_]:= If[n<1, 0, (-1/2)Pochhammer[-1/4, n] 8^n/n!] (* Michael Somos, Jan 17 2014 *)
a[ n_]:= SeriesCoefficient[(1 -(1-8x)^(1/4))/2, {x,0,n}] (* Michael Somos, Jan 17 2014 *)

{a(n) = if( n<0, 0, polcoeff( (1 - (1 - 8*x + x * O(x^n))^(1/4)) / 2, n))} /* Michael Somos, Jan 17 2014 */

[8^(n-1)*binomial(n-5/4,-1/4)/n for n in (1..40)] # G. C. Greubel, Aug 09 2022

a(n) = 2^(n-1)*3*7*11*...*(4n-5)/n! = 2*a(n-1)*(32*a(n-2) + a(n-1))/(18*a(n-2) -a(n-1)).
a(n) = -A004984(n)/2.
D-finite with recurrence n*a(n) + 2*(5-4*n)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012
G.f. A(x) =: y satisfies x = y * (1 - y) * (1 - 2*y + 2*y^2). - Michael Somos, Jan 17 2014
0 = a(n) * (64*a(n+1) - 18*a(n+2)) + a(n+1) * (2*a(n+1) + a(n+2)) unless n=0. - Michael Somos, Jan 17 2014
From Karol A. Penson, Dec 19 2015: (Start)
a(n) = 8^(n-1)*binomial(n-5/4, -1/4)/n.
E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [2], 8*x).
Representation as n-th moment of a positive function on (0, 8): a(n) = int(x^n*((2^(1/4)/(2*Pi*x^(1/4))*(1-x/8)^(1/4))), x=0..8), n=0,1,... . This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End)
a(n) ~ 2^(3*n-3) / (Gamma(3/4) * n^(5/4)). - Amiram Eldar, Sep 01 2025

A289391 Coefficients in expansion of E_14^(1/4).

Original entry on oeis.org

1, -6, -49212, -10451544, -4218246978, -1581565900392, -677142351901080, -293172823731286848, -132241381826055031692, -60651805300034501958126, -28350123351848675673466968, -13420046900399367136336144200

Offset: 0

Seiichi Manyama, Jul 05 2017

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

E_k^(1/4): A289392 (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), this sequence (k=14).

Cf. A004984, A058550 (E_14).

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[13, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A289029(n)/4).
a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3*Pi^2 / (2^(17/4) * Gamma(3/4)^9) = -0.2497407198517688195944362279691013167903920989625478927175764401875... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_13(k)*q^k. - Seiichi Manyama, Jun 16 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

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

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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

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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

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A289392 Coefficients in expansion of E_2^(1/4).

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A110150 G.f.: 4th root of Eisenstein series E_10 (cf. A013974).

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A048779 Coefficients of power series for (1 - (1-8*x)^(1/4))/2.

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A289391 Coefficients in expansion of E_14^(1/4).

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

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

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

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