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

0,4

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

References

B. Mazur, Perturbations, deformations and variations ..., Bull. Amer. Math. Soc., 41 (2004), 307-336.

Links

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

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

Example

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 + ...

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 *)

Prog

(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 */

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)).

Sequence in context: A299284 A348461 A002417 * A232772 A213776 A113751

Adjacent sequences: A126855 A126856 A126857 * A126859 A126860 A126861

Keyword

nonn

Author

N. J. A. Sloane, Mar 15 2007

Status

approved