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A005887 Theta series of f.c.c. lattice with respect to octahedral hole.
(Formerly M4070)
6

%I M4070 #32 Mar 12 2021 22:24:41

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

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

%H N. J. A. Sloane, <a href="/A005887/b005887.txt">Table of n, a(n) for n = 0..9999</a>

%H 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 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/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>

%F 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 A005875(2*n + 1) = a(n). - _Michael Somos_, Aug 17 2009

%e 6 + 8*x + 24*x^2 + 30*x^4 + 24*x^5 + 24*x^6 + 48*x^8 + 24*x^9 + 48*x^

%e 10 + ...

%e 6*q + 8*q^3 + 24*q^5 + 30*q^9 + 24*q^11 + 24*q^13 + 48*q^17 + 24*q^19 + ...

%p 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 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 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 (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 Cf. A005875.

%K nonn

%O 0,1

%A _N. J. A. Sloane_

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)