A294626 -id:A294626 - cp's OEIS frontend

A294974 Coefficients in expansion of (E_2^4/E_4)^(1/8).

1, -42, 4032, -659904, 118064226, -22406634432, 4407587356032, -888750999070464, 182478248639753472, -37986867560948245674, 7994272624037726124672, -1697243410477799687716416, 362963150140702802158191360, -78095916585903527021840348352

Offset: 0

Seiichi Manyama, Feb 12 2018

sign

Also coefficients in expansion of (E_2^8/E_8)^(1/16).

Cf. A004009 (E_4), A006352 (E_2), A108091, A289247, A289291, A294626, A294976.

terms = 14;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
(E2[x]^4/E4[x])^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: Product_{n>=1} (1-q^n)^A294626(n).

a(n) ~ (-1)^n * 2^(13/8) * Pi * exp(Pi*sqrt(3)*n) / (Gamma(1/8) * Gamma(1/3)^(9/4) * n^(7/8)). - Vaclav Kotesovec, Jun 03 2018

A294978 Coefficients in expansion of (E_4/E_2^4)^(1/8).

Original entry on oeis.org

1, 42, -2268, 395304, -64600914, 11644170552, -2188350306072, 424652412357696, -84326944950450972, 17044476557469661986, -3493525880987663047128, 724189608821718233434296, -151528575864988356484968840, 31955212589107172812017247992

Offset: 0

Seiichi Manyama, Feb 12 2018

sign

Also coefficients in expansion of (E_8/E_2^8)^(1/16).

Cf. A004009 (E_4), A006352 (E_2), A108091, A289565, A294626, A294974.

terms = 14;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
(E4[x]/E2[x]^4)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A294974.
G.f.: Product_{n>=1} (1-q^n)^(-A294626(n)).
a(n) ~ -(-1)^n * Pi^(5/4) * exp(Pi*sqrt(3)*n) / (2^(19/8) * 3^(9/8) * Gamma(2/3)^(9/4) * Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

A294974 Coefficients in expansion of (E_2^4/E_4)^(1/8).

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