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A045836 Half of Theta series of b.c.c. lattice with respect to long edge. 1
1, 2, 0, 0, 4, 4, 0, 0, 5, 4, 0, 0, 4, 8, 0, 0, 8, 6, 0, 0, 8, 4, 0, 0, 5, 12, 0, 0, 12, 8, 0, 0, 8, 8, 0, 0, 4, 12, 0, 0, 16, 8, 0, 0, 12, 8, 0, 0, 9, 14, 0, 0, 12, 16, 0, 0, 8, 4, 0, 0, 12, 16, 0, 0, 16, 16, 0, 0, 16, 8, 0, 0, 8, 20, 0, 0, 16, 8, 0, 0, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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. 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

LINKS

Table of n, a(n) for n=1..81.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Contribution from Michael Somos, May 31 2012: (Start)

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

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

Euler transform of period 8 sequence [ 2, -3, 2, 1, 2, -3, 2, -3, ...].

a(4*n) = a(4*n + 3) = 0. a(n) = 2 * A004025(n). a(4*n + 1) = A045834(n). a(4*n + 2) = 2 * A045828(n). (End)

EXAMPLE

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

MATHEMATICA

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[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-Fran├žois Alcover, Nov 05 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( 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. A004025, A045828, A045834.

Sequence in context: A004531 A072071 A329264 * A182056 A072070 A137830

Adjacent sequences:  A045833 A045834 A045835 * A045837 A045838 A045839

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)