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A213022 Expansion of phi(x)^2 * psi(x) in powers of x where phi(), psi() are Ramanujan theta functions. 8
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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-Fran├žois 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

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Last modified July 17 02:55 EDT 2019. Contains 325092 sequences. (Running on oeis4.)