A126858 Coefficients in quasimodular form F_2(q) of level 1 and weight 6.

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

Offset: 0

Author

N. J. A. Sloane, Mar 15 2007

Keywords

• nonn

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

Examples

			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 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Michael Andrew Henry, Hecke vector-forms: vector functions of quasiautomorphic forms over Hecke triangle groups, ResearchGate (2025). See p. 14.
• B. Mazur, Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.

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)).

Programs

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 *)

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 */

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