A289391 -id:A289391 - 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

sign

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

sign

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

A295817 Coefficients in expansion of E_14^(-1/4).

Original entry on oeis.org

1, 6, 49248, 11042304, 6770802642, 2705631701472, 1359219630420288, 633774007586896896, 312343963839774306864, 152751427857668869125990, 75972914003765783253275712, 37915118574439727639476081152, 19063775719322131645175269693920

Offset: 0

Seiichi Manyama, Feb 13 2018

nonn

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

Cf. A058550 (E_14), A289391.

Convolution inverse of A289391.

a(n) ~ 2^(7/4) * Gamma(3/4)^9 * exp(2*Pi*n) / (3 * Pi^3 * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A295817 Coefficients in expansion of E_14^(-1/4).

Views

Author

Keywords

Links

Crossrefs

Formula