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A290048 Coefficients in expansion of E_6*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

%I A290048 #28 Mar 11 2023 06:33:27
%S A290048 1,-552,8640,116000,-4868460,67855536,-544522240,2742137280,
%T A290048 -8237774250,10592091400,3366617856,113971542048,-1217020425880,
%U A290048 4535746506000,-5415752171520,-19090509870144,93580817811453,-142801363479240,-80721277168000,665065363025280
%N A290048 Coefficients in expansion of E_6*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).
%H A290048 Seiichi Manyama, <a href="/A290048/b290048.txt">Table of n, a(n) for n = 2..1000</a>
%H A290048 S. C. Milne, <a href="https://arxiv.org/abs/math/0009130">Hankel determinants of Eisenstein series</a>, preprint, arXiv:0009130 [math.NT], 2000.
%F A290048 Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_6, E_8, E_10 ; E_8, E_10, E_12 ; E_10, E_12, E_14]. G.f. is -691^2*b(q)/(1728^2*250^2).
%F A290048 a(n) = (A290050(n) - 2*691*A290049(n) + 691^2*A282382(n))/(1728^2*250^2).
%t A290048 terms = 20;
%t A290048 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A290048 E6[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)
%Y A290048 Cf. A000594, A010839, A013973 (E_6).
%Y A290048 Cf. A282382, A282461 (E_6*E_10*E_14 = E_10^3), A290049, A290050.
%Y A290048 E_k*Delta^2: A290178 (k=4), this sequence (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).
%K A290048 sign
%O A290048 2,2
%A A290048 _Seiichi Manyama_, Jul 19 2017