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%I M0928 #37 Jan 03 2023 14:50:08
%S 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,
%T 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,
%U 0,0,24,32,0,0,32,32,0,0,32,16,0,0,16,40,0,0,32
%N Theta series of b.c.c. lattice with respect to long edge.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A004025/b004025.txt">Table of n, a(n) for n = 1..1000</a>
%H N. J. A. Sloane and B. K. Teo, <a href="http://dx.doi.org/10.1063/1.449551">Theta series and magic numbers for close-packed spherical clusters</a>, J. Chem. Phys. 83 (1985) 6520-6534.
%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 <a href="/index/Ba#bcc">Index entries for sequences related to b.c.c. lattice</a>
%F From _Michael Somos_, May 31 2012: (Start)
%F 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.
%F Expansion of 2 * eta(q^2)^5 * eta(q^8)^4 / (eta(q)^2 * eta(q^4)^4) in powers of q.
%F 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)
%e 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 + ...
%t 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 *)
%o (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 */
%Y Cf. A045828, A045834, A045836.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
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