A213022 Expansion of phi(x)^2 * psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 5, 8, 5, 8, 16, 9, 8, 16, 8, 17, 24, 8, 16, 16, 13, 24, 16, 16, 24, 32, 13, 8, 32, 8, 24, 40, 16, 25, 24, 24, 24, 32, 16, 16, 40, 17, 32, 32, 16, 40, 48, 16, 16, 32, 21, 48, 32, 16, 24, 40, 32, 24, 56, 24, 45, 40, 16, 32, 24, 32, 40, 48, 16, 32, 64, 25, 24

Offset

0,2

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/8) * eta(q^2)^12 / (eta(q)^5 * eta(q^4)^4) in powers of q.

Expansion of q^(-1/16) times theta series of b.c.c. lattice with respect to point [0, 0, 1/4] in powers of q^(1/2).

Euler transform of period 4 sequence [ 5, -7, 5, -3, ...].

6 * a(n) = A005875(8*n + 1).

Example

a(0) = 1 since the norm squared of point [0, 0, 0] with respect to [0, 0, 1/4] is 1/16 = 1/16 + 1/2*0.
a(1) = 5 since the norm squared of points [-1/2, -1/2, -1/2], [-1/2, 1/2, -1/2], [0, 0, -1], [1/2, -1/2, -1/2], [1/2, 1/2, -1/2] with respect to [0, 0, 1/4] is 9/16 = 1/16 + 1/2*1.
1 + 5*x + 8*x^2 + 5*x^3 + 8*x^4 + 16*x^5 + 9*x^6 + 8*x^7 + 16*x^8 + 8*x^9 + ...
q + 5*q^9 + 8*q^17 + 5*q^25 + 8*q^33 + 16*q^41 + 9*q^49 + 8*q^57 + 16*q^65 + ...

Mathematica

CoefficientList[QPochhammer[q^2]^12/(QPochhammer[q]^5*QPochhammer[q^4]^4) + O[q]^70, q] (* Jean-François Alcover, Nov 05 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 / (eta(x + A)^5 * eta(x^4 + A)^4), n))}

Crossrefs

Cf. A005875.

Sequence in context: A101465 A010719 A246903 * A198732 A202348 A273817

Adjacent sequences: A213019 A213020 A213021 * A213023 A213024 A213025

Keyword

nonn

Author

Michael Somos, Jun 03 2012

Status

approved