A294183 -id:A294183 - cp's OEIS frontend

A294181 Coefficients in expansion of E_2/E_4.

1, -264, 61128, -14107296, 3255470952, -751247454384, 173361309784992, -40005651284526912, 9231887649122522280, -2130392752758423726312, 491619206548389935051568, -113448303808924351510423008, 26179851123971817380111236128

Offset: 0

Seiichi Manyama, Feb 11 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A001943, A004009 (E_4), A006352 (E_2), A288877.

E_k/E_{k+2}: this sequence (k=2), A294182 (k=4), A294183 (k=6).

terms = 13;
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]/E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Convolution inverse of A288877.

a(n) ~ (-1)^n * 1024 * Pi^11 * exp(Pi*sqrt(3)*n) / (3^(3/2) * Gamma(1/3)^18). - Vaclav Kotesovec, Jun 03 2018

A294182 Coefficients in expansion of E_4/E_6.

Original entry on oeis.org

1, 744, 393768, 210962976, 112966533672, 60492691156464, 32393330061359904, 17346357971979746880, 9288829947058862457384, 4974090926254339741926216, 2663584184830281769743846768, 1426327104764356980195826984032

Offset: 0

Seiichi Manyama, Feb 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A000706, A004009 (E_4), A013973 (E_6), A288261.

E_k/E_{k+2}: A294181 (k=2), this sequence (k=4), A294183 (k=6).

terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]/E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Convolution inverse of A288261.

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

cp's OEIS Frontend

A294181 Coefficients in expansion of E_2/E_4.

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