A217220 Theta series of Kagome net with respect to an atom.

1, 4, 0, 4, 6, 0, 0, 8, 0, 4, 0, 0, 6, 8, 0, 0, 6, 0, 0, 8, 0, 8, 0, 0, 0, 4, 0, 4, 12, 0, 0, 8, 0, 0, 0, 0, 6, 8, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 6, 12, 0, 0, 12, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 8, 6, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 4, 12, 0, 0, 8, 0, 4, 0, 0, 12, 0, 0, 0, 0, 0, 0, 16, 0, 8, 0, 0, 0, 8, 0, 0, 6, 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

Antti Karttunen, Table of n, a(n) for n = 0..65537

N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), pp. 1653-1657.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Phi_0(q)-phi_1(q^4) in the notation of SPLAG, Chapter 4.

a(n) = 4 * b(n) where b() is multiplicative with b(2^e) = (1+(-1)^e)*3/4, b(3^e) = 1, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p == 1 (mod 6). - Michael Somos, Feb 01 2017

Expansion of (2 * a(q) + a(q^4)) / 3 in powers of q where a() is a cubic AGM function. - Michael Somos, Feb 01 2017

Expansion of phi(q) * phi(q^3) + 2 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 01 2017

Example

G.f. = 1 + 4*q + 4*q^3 + 6*q^4 + 8*q^7 + 4*q^9 + 6*q^12 + 8*q^13 + ...

Maple

S:= series(JacobiTheta3(0, q)*JacobiTheta3(0, q^3)+JacobiTheta2(0, q)*JacobiTheta2(0, q^3)/2, q, 103):
seq(coeff(S, q, n), n=0..102); # Robert Israel, Nov 20 2017

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3] + 1/2 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}]; (* Michael Somos, Feb 01 2017 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 4 * sumdiv( n, d, kronecker( d, 3)) + if( n%4==0, 2 * sumdiv( n/4, d, kronecker( d, 3))))}; /* Michael Somos, Feb 01 2017 */
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 80); A[1] + 4*A[2] + 4*A[4] + 6*A[5]; /* Michael Somos, Feb 01 2017 */

Crossrefs

Cf. A217221.

Sequence in context: A258860 A134944 A088375 * A361620 A303560 A341481

Adjacent sequences: A217217 A217218 A217219 * A217221 A217222 A217223

Keyword

nonn

Author

N. J. A. Sloane, Oct 05 2012

Status

approved