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 A126858 Coefficients in quasimodular form F_2(q) of level 1 and weight 6. 5
 0, 0, 1, 8, 30, 80, 180, 336, 620, 960, 1590, 2200, 3416, 4368, 6440, 7920, 11160, 13056, 18333, 20520, 27860, 31360, 41052, 44528, 59760, 62400, 80990, 87120, 109872, 113680, 147960, 148800, 188976, 196416, 240210, 243040, 311910, 303696, 376580, 385840 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS This is also (5*E_2^3 - 3*E_2*E_4 - 2*E_6)/51840, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively. - N. J. A. Sloane, Feb 06 2017 This is also ((q*(d/dq)E_4)/240 + q*(d/dq)(q*(d/dq)E_2)/24)/6, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively. - Seiichi Manyama, Feb 08 2017 REFERENCES B. Mazur, Perturbations, deformations and variations ..., Bull. Amer. Math. Soc., 41 (2004), 307-336. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 FORMULA F_2(q) = (5*E(2)^3-3*E(2)*E(4)-2*E(6))/51840 where E(k) is the normalized Eisenstein series of weight k (cf. A006352, etc.). Expansion of (L1 * L2 - L3) / 720 where L1 = E2 (A006352), L2 = q * dL1/dq, L3 = q * dL2/dq in powers of q where E2 is an Eisenstein series. - Michael Somos, Jan 08 2012 a(n) = (A145094(n)/240 - A282154(n)/24)/6 = (A281372(n) - A282097(n))/6. - Seiichi Manyama, Feb 08 2017 EXAMPLE F_2(q) = q^2 + 8*q^3 + 30*q^4 + 80*q^5 + 180*q^6 + 336*q^7 + 620*q^8 + 960*q^9 + 1590*q^10 + 2200*q^11 + ... MAPLE with(numtheory); M:=100; E := proc(k) local n, t1; global M; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1); series(t1, q, M+1); end; e2:=E(2); e4:=E(4); e6:=E(6); series((5*e2^3-3*e2*e4-2*e6)/51840, q, M+1); seriestolist(%); # from N. J. A. Sloane, Feb 06 2017 MATHEMATICA terms = 40; Ei[n_] = 1 - (2 n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, 1, terms}]; S = 5 Ei[2]^3 - 3 Ei[2] Ei[4] - 2 Ei[6]; CoefficientList[S + O[x]^terms, x]/SeriesCoefficient[S, {x, 0, 2}] (* Jean-François Alcover, Feb 28 2018 *) PROG (PARI) {a(n) = local(L1, L2, L3); if( n<0, 0, L1 = 1 - 24 * sum( k = 1, n, sigma(k) * x^k, x * O(x^n)); L2 = x * L1'; L3 = x * L2'; polcoeff( (L1 * L2 - L3) / 720, n))} /* Michael Somos, Jan 08 2012 */ CROSSREFS Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145094 (q*(d/dq)E_4), A281372, A282097, A282154 (-q*(d/dq)(q*(d/dq)E_2)). Sequence in context: A063489 A299284 A002417 * A232772 A213776 A113751 Adjacent sequences:  A126855 A126856 A126857 * A126859 A126860 A126861 KEYWORD nonn AUTHOR N. J. A. Sloane, Mar 15 2007 STATUS approved

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Last modified October 14 05:08 EDT 2019. Contains 327995 sequences. (Running on oeis4.)