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 A288261 Coefficients in expansion of E_6/E_4. 19
 1, -744, 159768, -36866976, 8507424792, -1963211493744, 453039686271072, -104545516658693952, 24125403112135458840, -5567288717204029449672, 1284733088879405339418768, -296470902355240575283208928, 68414985730612787485819011168 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 5 17:46 EDT 2021. Contains 346488 sequences. (Running on oeis4.)