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A217221
Theta series of Kagome net with respect to a deep hole.
2
0, 6, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0
OFFSET
0,2
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
LINKS
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Phi_0(q)-phi_0(q^4) in the notation of SPLAG, Chapter 4.
Expansion of a(q) - a(q^4) in powers of q where a() is a cubic AGM function. - Michael Somos, Feb 01 2017
Expansion of 6 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 01 2017
Expansion of 6 * (eta(q^4) * eta(q^12))^2 / (eta(q^2) * eta(q^6)) in powers of q. - Michael Somos, Feb 01 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 27^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A115978. - Michael Somos, Feb 01 2017
a(2*n) = 0. a(2*n + 1) = 6 * A033762(n). - Michael Somos, Feb 01 2017
EXAMPLE
G.f. = 6*q + 6*q^3 + 12*q^7 + 6*q^9 + 12*q^13 + 12*q^19 + 12*q^21 + ...
MATHEMATICA
a[ n_] := If[ n < 1 || EvenQ[n], 0, 6 DivisorSum[n, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Feb 01 2017 *)
PROG
(PARI) {a(n) = if( n<1 || n%2==0, 0, 6 * sumdiv(n, d, kronecker(-3, d)))}; /* Michael Somos, Feb 01 2017 */
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 80); 6*A[2] + 6*A[4]; /* Michael Somos, Feb 01 2017 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 05 2012
STATUS
approved