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 A004013 Theta series of body-centered cubic (b.c.c.) lattice. (Formerly M4473) 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). 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). - Andrey Zabolotskiy, Mar 08 2020 REFERENCES 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). LINKS John Cannon, Table of n, a(n) for n = 0..10000 S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_4(q). R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013. G. Nebe and N. J. A. Sloane, Home page for this lattice Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Eric Weisstein's World of Mathematics, Theta Series FORMULA 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. a(8*n) = A004015(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). - Michael Somos, Jul 19 2015 a(12*n + 4) = 6 * A213056(n). a(16*n + 4) = 6 * A045834(n). a(16*n + 8) = 12 * A045828(n). EXAMPLE 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) + ... MAPLE 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: MATHEMATICA 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 *) PROG (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 */ CROSSREFS Cf. A004015, A005875, A008443, A045826, A045828, A045834, A213056. Sequence in context: A010116 A031365 A300382 * A010118 A226317 A100121 Adjacent sequences:  A004010 A004011 A004012 * A004014 A004015 A004016 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified January 28 04:10 EST 2021. Contains 340490 sequences. (Running on oeis4.)