A126859 Numerators of coefficients in quasimodular form F_3(q) of level 1 and weight 12.
0, 0, 1, 20, 102, 2288, 3773, 14232, 133616, 119904, 584517, 1927900, 4013432, 2569296, 14394518, 8365192, 14426496, 23381600, 151885575, 58125708, 269849564, 395149888, 195967551, 828880856, 398774464, 544543680, 4586626939, 1018905048, 1396485648
Offset: 0
Examples
F_3(q) = (1/12)*q^2 + (20/3)*q^3 + 102*q^4 + (2288/3)*q^5 + 3773*q^6 + 14232*q^7 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- B. Mazur, Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.
Programs
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Mathematica
es[2]=1-24*Sum[DivisorSigma[1, n]*q^n, {n, 100}]; es[k_?EvenQ]/; k>2:=1-2*k/BernoulliB[k]*Sum[DivisorSigma[k-1, n]*q^n, {n, 100}]; Numerator[CoefficientList[(15*es[2]^4*es[4] - 6*es[2]^6-12*es[2]^2*es[4]^2 + 7*es[4]^3 + 4*es[2]^3*es[6] - 12*es[2]*es[4]*es[6] + 4*es[6]^2)/35831808, q]][[;; 30]] (* Shenghui Yang, Aug 06 2025 *)
Formula
F_3(q) = (15*E(2)^4*E(4) - 6*E(2)^6 - 12*E(2)^2*E(4)^2 + 7*E(4)^3 + 4*E(2)^3*E(6) - 12*E(2)*E(4)*E(6) + 4*E(6)^2)/35831808, where E(k) is the normalized Eisenstein series of weight k (cf. A006352, etc.).