A288261 Coefficients in expansion of E_6/E_4.

1, -744, 159768, -36866976, 8507424792, -1963211493744, 453039686271072, -104545516658693952, 24125403112135458840, -5567288717204029449672, 1284733088879405339418768, -296470902355240575283208928, 68414985730612787485819011168

Offset

0,2

Comments

Also coefficients in expansion of E_10/E_8. - Seiichi Manyama, Jun 20 2017

Links

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

Formula

From Seiichi Manyama, Jun 27 2017: (Start)

Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n((-1+sqrt(3)*i)/2).

G.f.: Sum_{n >= 0} j_n((-1+sqrt(3)*i)/2)*q^n. (End)

a(n) ~ (-1)^n * 3 * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jun 28 2017

G.f.: -q*j'/j where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

Example

G.f.: 1 - 744*q + 159768*q^2 - 36866976*q^3 + 8507424792*q^4 - 1963211493744*q^5 + 453039686271072*q^6 + ...
From Seiichi Manyama, Jun 27 2017: (Start)
a(0) = j_0((-1+sqrt(3)*i)/2) = 1,_
a(1) = j_1((-1+sqrt(3)*i)/2) = -744 + 0^1 = -744,
a(2) = j_2((-1+sqrt(3)*i)/2) = 159768 - 1488*0^1 + 0^2 = 159768. (End)

Mathematica

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 13; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[6]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
a[ n_] := With[{j = Series[1728 KleinInvariantJ[ Log[ Series[q, {q, 0, n + 1}]]/(2 Pi I)], {q, 0, n}]}, SeriesCoefficient[ -q D[j, q] / j, {q, 0, n}]]; (* Michael Somos, Aug 15 2018 *)

Crossrefs

Cf. A004009 (E_4), A110163, A013973 (E_6).

E_{k+2}/E_k: A288877 (k=2), this sequence (k=4, 8), A288840 (k=6).

Cf. A000521 (j), A035230 (-q*j'), A066395 (1/j), A289141.

Sequence in context: A306281 A210178 A192731 * A000521 A178449 A178451

Adjacent sequences: A288258 A288259 A288260 * A288262 A288263 A288264

Keyword

sign

Author

Seiichi Manyama, Jun 17 2017

Status

approved