%I M2195 #23 Feb 16 2025 08:32:29

%S 0,0,0,3,0,0,1,0,0,3,0,1,2,0,0,4,0,2,2,2,2,2,1,2,1,1,0,4,0,0,0,2,1,6,

%T 2,4,1,2,1,2,0,5,2,3,1,6,0,4,0,4,2,2,2,4,0,2,0,5,2,2,2,4,0,2,1,4,3,5,

%U 2,2,0,2,2,9,2,6,3,6,0,4,2,2,3,8,2,2,1

%N Theta series of hexagonal close-packing with respect to center of triangle between two layers.

%C The triangle separates a tetrahedron and an octahedron.

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

%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="/A005890/b005890.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane and B. K. Teo, <a href="https://doi.org/10.1063/1.449551">Theta series and magic numbers for close-packed spherical clusters</a>, J. Chem. Phys. 83 (1985) 6520-6534. See page 6530 equation 66.

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

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

%F Expansion of x^3 * ( f(x^3, x^15) * (f(x^16, x^32) * f(x^15, x^39) + x^6 * f(x^8, x^40) * f(x^3, x^51)) + f(x^6, x^12) * (f(x^16, x^32) * f(x^12, x^42) + f(x^8, x^40) * f(x^24, x^30)) ) in powers of x where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Feb 11 2018

%F G.f.: Sum{i, j, k in Z} x^(9*(i*i + i*j + j*j) + 24*k*k) * (x^(6 - 12*(i+j) - 8*k) + x^(3 - 3*(i+j) + 16*k)). - _Michael Somos_, Feb 11 2018

%e G.f. = 3*x^3 + x^6 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + 2*x^17 + 2*x^18 + 2*x^19 + ...

%t f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a[n_] := SeriesCoefficient[x^3*(f[x^3, x^15]*(f[x^16, x^32]* f[x^15, x^39] + x^6*f[x^8, x^40]*f[x^3, x^51]) + f[x^6, x^12]*(f[x^16, x^32]*f[x^12, x^42] + f[x^8, x^40]*f[x^24, x^30])), {x, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Apr 02 2018 *)

%Y Cf. A004012, A005872, A005873, A005874, A005889.

%K nonn,changed

%O 0,4

%A _N. J. A. Sloane_