

A004025


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


2



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
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OFFSET

1,1


COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The bodycentered 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 closepacked spherical clusters, J. Chem. Phys. 83 (1985) 65206534.
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) = 1/2 * 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}] (* JeanFrançois 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: A230423 A213672 A309244 * A102561 A072068 A078145
Adjacent sequences: A004022 A004023 A004024 * A004026 A004027 A004028


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



