A004024 Theta series of b.c.c. lattice with respect to deep hole. (Formerly M3227)

4, 4, 8, 12, 4, 12, 12, 12, 16, 16, 8, 8, 28, 12, 20, 24, 8, 16, 28, 12, 16, 28, 20, 32, 20, 16, 16, 32, 20, 24, 28, 8, 36, 44, 12, 32, 36, 16, 24, 20, 28, 20, 56, 28, 16, 40, 20, 40, 44, 12, 36, 40, 20, 32, 40, 16, 24, 60, 32, 36, 40, 24, 32, 60, 24, 40, 24, 20, 60, 36, 24, 32, 56, 32

Offset

0,1

References

Ono and Skinner, Ann. Math., 147 (1998), 453-470.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Index entries for sequences related to b.c.c. lattice

Formula

4*eta(32z)^4/eta(8z) = 4*Sum q^(x^2+2y^2+2z^2), x, y, z >= 1 and odd.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(5*Pi/8) * Pi^(3/4) * 2^(7/8) / Gamma(3/4)^3 = A388117. - Simon Plouffe, Sep 15 2025

Mathematica

max = 73; 4*CoefficientList[ Series[ Product[ (1-q^(4k))^4 / (1-q^k), {k, 1, max}], {q, 0, max}], q] (* Jean-François Alcover, Feb 10 2012, after A045831 *)
terms = 74; QP = QPochhammer; s = 4 QP[z^4]^4/QP[z] + O[z]^terms; CoefficientList[s, z] (* Jean-François Alcover, Jul 07 2017 *)

Crossrefs

Equals 4*A045831.

Sequence in context: A376433 A301705 A309456 * A292276 A278083 A086663

Adjacent sequences: A004021 A004022 A004023 * A004025 A004026 A004027

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved