A004025 Theta series of b.c.c. lattice with respect to long edge. (Formerly M0928)

2, 4, 0, 0, 8, 8, 0, 0, 10, 8, 0, 0, 8, 16, 0, 0, 16, 12, 0, 0, 16, 8, 0, 0, 10, 24, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 8, 24, 0, 0, 32, 16, 0, 0, 24, 16, 0, 0, 18, 28, 0, 0, 24, 32, 0, 0, 16, 8, 0, 0, 24, 32, 0, 0, 32, 32, 0, 0, 32, 16, 0, 0, 16, 40, 0, 0, 32

Offset

1,1

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The body-centered cubic (b.c.c. also known as D3*) lattice is the set of all triples [a, b, c] where the entries are all integers or all one half an odd integer. A long edge is centered at a triple with two integer entries and the remaining entry is one half an odd integer. - Michael Somos, May 31 2012

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Formula

From Michael Somos, May 31 2012: (Start)

Expansion of 2 * x * phi(x) * psi(x^4)^2 = 2 * x * psi(-x^2)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of 2 * eta(q^2)^5 * eta(q^8)^4 / (eta(q)^2 * eta(q^4)^4) in powers of q.

a(4*n) = a(4*n + 3) = 0. a(n) = 2 * A045836(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = 4 * A045828(n). (End)

Example

2*q + 4*q^2 + 8*q^5 + 8*q^6 + 10*q^9 + 8*q^10 + 8*q^13 + 16*q^14 + 16*q^17 + ...

Mathematica

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[2*QPochhammer[x^2+A]^5 * (QPochhammer[x^8+A]^4 / (QPochhammer[x+A]^2*QPochhammer[x^4+A]^4)), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^4 / (eta(x + A)^2 * eta(x^4 + A)^4), n))} /* Michael Somos, May 31 2012 */

Crossrefs

Cf. A045828, A045834, A045836.

Sequence in context: A213672 A309244 A378518 * A102561 A072068 A078145

Adjacent sequences: A004022 A004023 A004024 * A004026 A004027 A004028

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved