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A004025
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Theta series of b.c.c. lattice with respect to long edge.
(Formerly M0928)
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2
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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
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1,1
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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)
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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