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A005887 Theta series of f.c.c. lattice with respect to octahedral hole.

%I A005887 M4070 #33 Feb 16 2025 08:32:29
%S A005887 6,8,24,0,30,24,24,0,48,24,48,0,30,32,72,0,48,48,24,0,96,24,72,0,54,
%T A005887 48,72,0,48,72,72,0,96,24,96,0,48,56,96,0,102,72,48,0,144,48,48,0,48,
%U A005887 72,168,0,96,72,72,0,96,48,120,0,78,48,144,0,144,120,48,0,96,72,96,0,96,56,168
%N A005887 Theta series of f.c.c. lattice with respect to octahedral hole.
%C A005887 Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%D A005887 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005887 N. J. A. Sloane, <a href="/A005887/b005887.txt">Table of n, a(n) for n = 0..9999</a>
%H A005887 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html">Home page for this lattice</a>
%H A005887 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 A005887 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A005887 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A005887 <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>
%F A005887 Expansion of q^(-1) * (phi^3(q) - phi^3(-q)) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - _Michael Somos_, Aug 17 2009
%F A005887 A005875(2*n + 1) = a(n). - _Michael Somos_, Aug 17 2009
%e A005887 6 + 8*x + 24*x^2 + 30*x^4 + 24*x^5 + 24*x^6 + 48*x^8 + 24*x^9 + 48*x^
%e A005887 10 + ...
%e A005887 6*q + 8*q^3 + 24*q^5 + 30*q^9 + 24*q^11 + 24*q^13 + 48*q^17 + 24*q^19 + ...
%p A005887 maxd:=20001: read format: temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a,q,maxd): a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd): th4:=series(subs(q=-q,th3),q,maxd):
%p A005887 t1:=series((th3^3-th4^3)/(2*q),q,maxd): t1:=series(subs(q=sqrt(q),t1),q,floor(maxd/2)): t2:=seriestolist(t1): for n from 1 to nops(t2) do lprint(n-1, t2[n]); od:
%t A005887 s = (EllipticTheta[3, 0, q]^3 - EllipticTheta[3, 0, -q]^3)/(2q) + O[q]^200; CoefficientList[s, q^2] (* _Jean-François Alcover_, Sep 19 2016 *)
%o A005887 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n))^3, n))} /* _Michael Somos_, Aug 17 2009 */
%Y A005887 Cf. A005875.
%K A005887 nonn
%O A005887 0,1
%A A005887 _N. J. A. Sloane_