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A004013
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Theta series of body-centered cubic (b.c.c.) lattice.
(Formerly M4473)
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8
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1, 0, 0, 8, 6, 0, 0, 0, 12, 0, 0, 24, 8, 0, 0, 0, 6, 0, 0, 24, 24, 0, 0, 0, 24, 0, 0, 32, 0, 0, 0, 0, 12, 0, 0, 48, 30, 0, 0, 0, 24, 0, 0, 24, 24, 0, 0, 0, 8, 0, 0, 48, 24, 0, 0, 0, 48, 0, 0, 72, 0, 0, 0, 0, 6, 0, 0, 24, 48, 0, 0, 0, 36, 0, 0, 56, 24, 0, 0, 0, 24, 0, 0, 72, 48, 0, 0, 0, 24, 0, 0
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OFFSET
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0,4
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COMMENTS
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The bcc lattice is also called the coweight lattice A^*_3 (or D^*_3), dual to the root lattice A_3 (or D_3, or the fcc lattice), or the permutohedral lattice A^*_3. - Andrey Zabolotskiy, Mar 08 2020
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116.
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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subs(q=q^2, ph)^3+(2*sqrt(q))^3*subs(q=q^4, ps)^3, where ps = A010054 = Sum_{k=0..infinity} q^(k*(k+1)/2), ph = A000122 = Sum_{k=-infinity, infinity} q^(k^2).
Expansion of phi(q^4)^3 + 8 * q^3 * psi(q^8)^3 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A005875(n).
Expansion of theta_3(q)^3 + theta_2(q)^3 in powers of q^(1/4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004015.
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EXAMPLE
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G.f. = 1 + 8*x^3 + 6*x^4 + 12*x^8 + 24*x^11 + 8*x^12 + 6*x^16 + 24*x^19 + 24*x^20 + ...
G.f. = 1 + 8*q^(3/2) + 6*q^2 + 12*q^4 + 24*q^(11/2) + 8*q^6 + 6*q^8 + 24*q^(19/2) + 24*q^10 + 24*q^12 + 32*q^(27/2) + ...
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MAPLE
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M:=100; M1:=M*(M+1)/2; ph:=series(add(q^(k^2), k=-M..M), q, M1): ps:=series(add(q^(k*(k+1)/2), k=0..M), q, M1): t1:=series(subs(q=q^2, ph)^3, q, M1): t2:=series((2*sqrt(q))^3*subs(q=q^4, ps)^3, q, M1): t3:=seriestolist(series(subs(q=q^2, t1+t2), q, M1)): for n from 0 to nops(t3)-1 do lprint(n, t3[n+1]); od:
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MATHEMATICA
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m = 13; m1 = m*((m + 1)/2); ph[q_] = Series[ Sum[ q^k^2, {k, -m, m}], {q, 0, m1}]; ps[q_] = Series[ Sum[ q^(k*((k + 1)/2)), {k, 0, m}], {q, 0, m1}]; t1[q_] = Normal[ Series[ ph[q^2]^3, {q, 0, m1}]]; t2[q_] = Normal[ Series[ (2*Sqrt[q])^3*ps[q^4]^3, {q, 0, m1}]]; CoefficientList[ Series[ t1[q^2] + t2[q^2], {q, 0, m1}], q] (* Jean-François Alcover, Dec 20 2011, translated from Maple *)
(* From version 6 on *) terms=91; f[q_] = LatticeData["BodyCenteredCubic", "ThetaSeriesFunction"][-I Log[q]/Pi]; CoefficientList[Simplify[f[q] + O[q]^terms, q>0], q][[1 ;; terms]] (* Jean-François Alcover, May 15 2013, updated Jul 08 2017 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^4]^3 + EllipticTheta[ 2, 0, x^4]^3, {x, 0, n}]; (* Michael Somos, May 24 2013 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, if( n%4==0, n/=4; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^3, n), n%8==3, n\=8; 8*polcoeff( sum(k=0, (sqrtint(8*n+1) - 1)\2, x^((k^2 + k)/2), x * O(x^n))^3, n)))}; /* Michael Somos, Oct 25 2006 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A)^5 / eta(x^4 + A)^2 / eta(x^16 + A)^2)^3 + (2 * x * eta(x^16 + A)^2 / eta(x^8 + A))^3, n))}; /* Michael Somos, May 17 2008 */
(Magma) Basis( ModularForms( Gamma0(8), 3/2), 90) [1]; /* Michael Somos, Sep 04 2014 */
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CROSSREFS
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Indices of nonzero terms are A004014.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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