A110163 Exponents a(1), a(2), ... such that theta series of E_8 lattice, 1 + 240 q + 2160 q^2 + ... (A004009) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ...

-240, 26760, -4096240, 708938760, -130880766192, 25168873498760, -4978357936128240, 1005225129317834760, -206195878414962688240, 42824436296045618358408, -8983966738037593190400240, 1900416270294787067711818760, -404814256845771786255876096240, 86744167089111545378556727322760

Offset: 1

Author

N. J. A. Sloane, Sep 16 2005

Keywords

• sign

Comments

Negative of inverse Euler transform of [240, 2160, ...].

Examples

			From _Seiichi Manyama_, Jun 17 2017: (Start)
a(1) = 8 + 1/3 * A008683(1/1) * A288261(1) = 8 + 1/3 * (-744) = -240,
a(2) = 8 + 1/6 * (A008683(2/1) * A288261(1) + A008683(2/2) * A288261(2)) = 8 + 1/6 * (744 + 159768) = 26760. (End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..424

Crossrefs

Cf. A288968 (k=2), this sequence (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A004009, A008683, A013953, A288261, A289636.

Programs

Mathematica

terms = 14; Clear[a, sol];
a4009[n_] := If[n == 0, 1, 240 DivisorSigma[3, n]];
sol[0] = {}; sol[kmax_] := sol[kmax] = Join[sol[kmax-1], SolveAlways[ Sum[ a4009[k] q^k, {k, 0, kmax}] == Normal[Product[(1-q^k)^a[k], {k, 1, kmax}] + O[q]^(kmax+1)] /. sol[kmax-1], q][[1]] ];
A110163 = Array[a, terms] /. sol[terms] (* Jean-François Alcover, Jul 03 2017 *)

Formula

a(n) = A013953(n^2) for n>=1. - Seiichi Manyama, Jun 17 2017

a(n) = 8 + (1/(3*n)) * Sum_{d|n} A008683(n/d) * A288261(d). - Seiichi Manyama, Jun 17 2017

a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289636(d). - Seiichi Manyama, Jul 09 2017

a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 08 2018