A004024 Theta series of b.c.c. lattice with respect to deep hole.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

Equals 4*A045831.

Programs

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 *)

Formula

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