A294975 -id:A294975 - cp's OEIS frontend

A294976 Coefficients in expansion of (E_6/E_2^6)^(1/12).

1, -30, -11340, -3912600, -1520905170, -636170644008, -278687199310200, -126000360658968000, -58290111778749466140, -27440829122946510954630, -13096614404248661886145848, -6320198941502349713305002120, -3077986352751848627729986859400

Offset: 0

Seiichi Manyama, Feb 12 2018

sign

Cf. A109817, A289565, A294974, A294975.

terms = 13;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]/E2[x]^6)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: Product_{n>=1} (1-q^n)^A294975(n).
a(n) ~ -Gamma(1/3)^2 * Gamma(1/4)^(10/3) * exp(2*Pi*n) / (16 * 2^(1/12) * 3^(7/12) * Pi^(5/2) * Gamma(1/12) * n^(13/12)). - Vaclav Kotesovec, Jun 03 2018
Equivalently, a(n) ~ -Gamma(1/3) * Gamma(1/4)^(7/3) * exp(2*Pi*n) / (2^(23/6) * 3^(23/24) * Pi^2 * sqrt(1 + sqrt(3)) * n^(13/12)). - Vaclav Kotesovec, Nov 26 2024

A294979 Coefficients in expansion of (E_2^6/E_6)^(1/12).

Original entry on oeis.org

1, 30, 12240, 4620000, 1915684770, 839549366208, 381374756189280, 177631327935911040, 84272487587664762240, 40549569894460426101150, 19730577674798681251391712, 9687875889040210133058857760, 4792614349874614536514510456320

Offset: 0

Seiichi Manyama, Feb 12 2018

nonn

Cf. A289291, A289540, A294975, A294976.

terms = 13;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E2[x]^6/E6[x])^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A294976.
G.f.: Product_{n>=1} (1-q^n)^(-A294975(n)).
a(n) ~ 2^(13/12) * 3^(1/3) * sqrt(Pi) * exp(2*Pi*n) / (Gamma(1/12) * Gamma(1/4)^(4/3) * n^(11/12)). - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

A294976 Coefficients in expansion of (E_6/E_2^6)^(1/12).

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