A282154 Coefficients in expansion of Eisenstein series -q*(d/dq)(q*(d/dq)E_2).

0, 24, 288, 864, 2688, 3600, 10368, 9408, 23040, 25272, 43200, 34848, 96768, 56784, 112896, 129600, 190464, 124848, 303264, 173280, 403200, 338688, 418176, 304704, 829440, 465000, 681408, 699840, 1053696, 605520, 1555200, 738048, 1548288, 1254528, 1498176

Offset: 0

Author

Seiichi Manyama, Feb 07 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006352 (E_2), A076835 (-q*(d/dq)E_2), this sequence (-q*(d/dq)(q*(d/dq)E_2)).

Cf. A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4), A282097.

This sequence is related to A126858.

Programs

Mathematica

terms = 35;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
-x*D[x*D[E2[x], x], x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

Formula

-q*(d/dq)(q*(d/dq)E_2) = -q*(d/dq)((E_2^2 - E_4)/12) = -(E_2^3 - 3*E_2*E_4 + 2*E_6)/72.

a(n) = -(A282018(n) - 3*A282019(n) + 2*A013973(n))/72.

a(n) = 24*A282097(n).