A109772 -id:A109772 - cp's OEIS frontend

A008427 Theta series of {D_8}* lattice.

1, 16, 368, 448, 3184, 2016, 10304, 5504, 25712, 12112, 46368, 21312, 89152, 35168, 126592, 56448, 205936, 78624, 278576, 109760, 401184, 154112, 490176, 194688, 719936, 252016, 808864, 327040

Offset: 0

N. J. A. Sloane

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

G. C. Greubel, Table of n, a(n) for n = 0..10000
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
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.
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A008430, A109772.

terms = 28; s = EllipticTheta[3, 0, q]^8 + EllipticTheta[2, 0, q]^8 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017 *)

G.f.: (theta_3(q))^8 + (theta_2(q))^8.

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

Original entry on oeis.org

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

Offset: 0

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

sign

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.

Cf. A008430, A008427, A109772.

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 *)

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

cp's OEIS Frontend

A008427 Theta series of {D_8}* lattice.

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