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%I M4040 #15 Jul 06 2018 20:44:42
%S 0,0,0,6,0,0,0,0,0,6,0,6,0,0,0,12,0,6,0,0,0,0,0,12,0,0,0,18,0,0,0,0,0,
%T 12,0,12,0,0,0,24,0,6,0,0,0,0,0,12,0,0,0,24,0,0,0,0,0,24,0,6,0,0,0,36,
%U 0,12,0,0,0,0,0,12,0,0,0,30,0,0,0,0,0,18
%N Theta series of hexagonal close-packing with respect to octahedral hole.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
%H Sean A. Irvine, <a href="/A005872/b005872.txt">Table of n, a(n) for n = 0..999</a>
%F a(2*n) = 0. a(2*n + 3) = 6*A298931(n). - _Michael Somos_, Jul 06 2018
%e G.f. = 6*q^3 + 6*q^9 + 6*q^11 + 12*q^15 + 6*q^17 + 12*q^23 + 18*q^27 + ... - _Michael Somos_, Jul 06 2018
%t a[ n_] := SeriesCoefficient[ 6 x^3 QPochhammer[ x^16]^2 QPochhammer[ x^18]^3 / (QPochhammer[ x^6] QPochhammer[ x^8]), {x, 0, n}]; (* _Michael Somos_, Jul 06 2018 *)
%o (PARI) {a(n) = my(A, m); if( n<3 || n%2==0, 0, m = n\2 - 1; A = x * O(x^m); 6 * polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), m))}; /* _Michael Somos_, Jul 06 2018 */
%Y Cf. A298931.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_.
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