A004013 - OEIS

%I M4473 #57 Sep 08 2022 08:44:32

%S 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,

%T 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,

%U 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

%N Theta series of body-centered cubic (b.c.c.) lattice.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C 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

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H John Cannon, <a href="/A004013/b004013.txt">Table of n, a(n) for n = 0..10000</a>

%H S. Ahlgren, <a href="http://dx.doi.org/10.1090/S0002-9939-99-05181-3">The sixth, eighth, ninth and tenth powers of Ramanujan's theta function</a>, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_4(q).

%H R. J. Mathar, <a href="http://arxiv.org/abs/1309.3705">Hierarchical Subdivision of the Simple Cubic Lattice</a>, arXiv preprint arXiv:1309.3705 [math.MG], 2013.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds3.html">Home page for this lattice</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ThetaSeries.html">Theta Series</a>

%H <a href="/index/Ba#bcc">Index entries for sequences related to b.c.c. lattice</a>

%F 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).

%F 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

%F a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A005875(n).

%F Expansion of theta_3(q)^3 + theta_2(q)^3 in powers of q^(1/4).

%F 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.

%F a(8*n) = A004015(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). - _Michael Somos_, Jul 19 2015

%F a(12*n + 4) = 6 * A213056(n). a(16*n + 4) = 6 * A045834(n). a(16*n + 8) = 12 * A045828(n).

%e 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 + ...

%e 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) + ...

%p 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:

%t 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 *)

%t (* 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 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^4]^3 + EllipticTheta[ 2, 0, x^4]^3, {x, 0, n}]; (* _Michael Somos_, May 24 2013 *)

%o (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 */

%o (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 */

%o (Magma) Basis( ModularForms( Gamma0(8), 3/2), 90) [1]; /* _Michael Somos_, Sep 04 2014 */

%Y Cf. A004015, A005875, A008443, A045826, A045828, A045834, A213056.

%Y Indices of nonzero terms are A004014.

%Y Cf. A023916-A023936.

%Y Cf. A004011, A008422, A008425, A008423, A008427, A008424, A008426.

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_