A217513 -id:A217513 - cp's OEIS frontend

A005925 Theta series of diamond.

1, 0, 0, 4, 0, 0, 0, 0, 12, 0, 0, 12, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 24, 0, 0, 16, 0, 0, 0, 0, 12, 0, 0, 24, 0, 0, 0, 0, 24, 0, 0, 12, 0, 0, 0, 0, 8, 0, 0, 24, 0, 0, 0, 0, 48, 0, 0, 36, 0, 0, 0, 0, 6, 0, 0, 12

Offset: 0

N. J. A. Sloane

a(n) > 0 iff n is in A047470. - Robert Israel, Jul 06 2016

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

Robert Israel, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 120.
G. L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From _N. J. A. Sloane_, Dec 18 2012
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A005926, A005927, A047470, A217512, A217513, A217514.

S:= series((JacobiTheta2(0,z^4)^3 + JacobiTheta3(0,z^4)^3 + JacobiTheta4(0,z^4)^3)/2, z, 101):
seq(coeff(S,z,j),j=0..100); # Robert Israel, Jul 06 2016

terms = 68; s = Simplify[Normal[(EllipticTheta[2, 0, z^4]^3 + EllipticTheta[3, 0, z^4]^3 + EllipticTheta[4, 0, z^4]^3)/2 + O[z]^terms], z > 0]; CoefficientList[s, z] (* Jean-François Alcover, Jul 07 2017 *)

(theta_2^3 + theta_3^3 + theta_4^3) / 2.

A005926 Theta series of diamond with respect to midpoint of edge.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0

Offset: 0

N. J. A. Sloane

			G.f. = 2*q^(3/16) + 6*q^(19/16) + 12*q^(35/16) + 12*q^(51/16) + 6*q^(67/16) + 18*q^(83/16) + 18*q^(99/16) + ...

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

Andy Huchala, Table of n, a(n) for n = 0..1600 (first 387 terms from Herman Jamke)
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A045840, A217513.

prec = 10;
eta[q_, a_] := Sum[q^((i + a)^2), {i, Range[-prec, prec]}];
t2[q_] := eta[q, 1/2];
t3[q_] := eta[q, 0];
T = Expand[t2[q^(1/2)]*(t2[q^2]*eta[q^4, 3/8] + t3[q^2]*eta[q^4, 1/8])] // PowerExpand;
A = Range[prec*16 + 1];
Do[A[[i + 1]] = Coefficient[T, q, i/16], {i, 1, prec*16}];
A[[1]] = 0; A  (* Andy Huchala, May 17 2023 *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

cp's OEIS Frontend

A005925 Theta series of diamond.

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