A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504*q - 16632*q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

504, 143388, 51180024, 20556578700, 8806299845112, 3929750661380124, 1803727445909594616, 845145871847732769804, 402283166289266872824312, 193877350835487271784566812, 94381548697864188120110027256, 46328820782943001597184984563596

Offset: 1

Author

Seiichi Manyama, Jun 18 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..367
• R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

Crossrefs

Cf. A288968 (k=2), A110163 (k=4), this sequence (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A013973 (E_6), A110163, A288840 (E_8/E_6), A289637.

Formula

a(n) = A013975(n^2) for n>=1.

a(n) = 12 + (1/(2*n)) * Sum_{d|n} A008683(n/d) * A288840(d).

a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289637(d). - Seiichi Manyama, Jul 09 2017

a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018