A109773 Expansion of eighth root of theta series of D_8 lattice.

1, 14, -544, 34496, -2512254, 197053696, -16194254272, 1374326128896, -119403428951808, 10561444878559246, -947458249960057024, 85971010094510200128, -7874673015172093889024, 727016151987267244001536, -67573426491012510177925760, 6317185611058637805840976640

Offset: 0

Author

N. J. A. Sloane and Nadia Heninger, Aug 13 2005

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Crossrefs

Cf. A008430, A008427, A109772.

Programs

Mathematica

terms = 16; f[q_] = LatticeData["D8", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q]^(1/8), {q, 0, 2 terms}]; CoefficientList[s, q^2][[1 ;; terms]] // Round (* Jean-François Alcover, Jul 07 2017 *)
CoefficientList[Series[((EllipticTheta[3, 0, Sqrt[x]]^8 + EllipticTheta[4, 0, Sqrt[x]]^8)/2)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 10 2017 *)

Formula

a(n) ~ -(-1)^n * c * d^n / n^(9/8), where d = -1/EllipticNomeQ(-3+2*sqrt(2)) = 101.05698591144255836558034070124390358691255555299851256465840129800034600429... and c = 0.11312909975079493828483346745366595624358550348529207605517154972... - Vaclav Kotesovec, Dec 11 2017, updated Mar 16 2024