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

1, -552, 8640, 116000, -4868460, 67855536, -544522240, 2742137280, -8237774250, 10592091400, 3366617856, 113971542048, -1217020425880, 4535746506000, -5415752171520, -19090509870144, 93580817811453, -142801363479240, -80721277168000, 665065363025280

Offset: 2

Author

Seiichi Manyama, Jul 19 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 2..1000
• S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

Crossrefs

Cf. A000594, A010839, A013973 (E_6).

Cf. A282382, A282461 (E_6*E_10*E_14 = E_10^3), A290049, A290050.

E_k*Delta^2: A290178 (k=4), this sequence (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).

Programs

Mathematica

terms = 20;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

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).

a(n) = (A290050(n) - 2*691*A290049(n) + 691^2*A282382(n))/(1728^2*250^2).