A280869 Expansion of E_6(q)^2 in powers of q.

1, -1008, 220752, 16519104, 399517776, 4624512480, 34423752384, 187506813312, 814794618960, 2975666040144, 9486668147040, 27052407031104, 70486610910912, 169931677686624, 384163181281152, 820165393918080, 1668889095288912, 3249638073414432

Offset: 0

Author

Seiichi Manyama, Jan 28 2017

Keywords

• sign

Examples

			G.f. = 1 - 1008*q + 220752*q^2 + 16519104*q^3 + 399517776*q^4 + 4624512480*q^5 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Eisenstein Series.

Crossrefs

Cf. A000594, A001160, A008411, A013973 (E_6), A029828 (691*E_12).

Programs

Mathematica

terms = 18;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

E6(q)^2 = (1 - 504 Sum_{i>=1} sigma_5(i)q^i)^2 where sigma_5(n) is A001160.

A008411(n) - a(n) = 1728*A000594(n).

A029828(n) - 691*a(n) = 762048*A000594(n).

A029828(n) = 441*A008411(n) + 250*a(n).