A002706 Theta series of 6-dimensional lattice A_6^(2) (other names for this lattice or the corresponding quadratic form are LAMBDA_{3,lambda}, P_6^(5), phi_6, F_14).

1, 0, 42, 56, 84, 168, 280, 336, 462, 336, 840, 672, 1176, 1176, 1386, 1008, 1848, 2016, 2058, 2520, 3528, 2408, 3108, 2688, 4760, 3024, 5880, 4592, 6468, 4704, 5040, 6720, 6930, 6832, 10080, 7224, 7812, 7392, 12600, 7056, 14280, 11760, 12040, 9408

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In Elkies 1999 the g.f. is denoted by theta_L. - Michael Somos, Nov 09 2014

Examples

			G.f. = 1 + 42*q^2 + 56*q^3 + 84*q^4 + 168*q^5 + 280*q^6 + 336*q^7 + 462*q^8 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Intro. to 3rd ed.
• N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103). See page 72.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• J. H. Conway and N. J. A. Sloane, Complex and integral laminated lattices, Trans. Amer. Math. Soc., 280 (1983), 463-490.
• N. Elkies, The Klein quartic in number theory
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Cf. A002653, A002656.

Programs

Magma

A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 42*A[3] + 56*A[4] + 84*A[5] + 168*A[6] + 280*A[7];  /* Michael Somos, Nov 09 2014 */

Mathematica

s = (EllipticTheta[3, 0, q] *EllipticTheta[3, 0, q^7] + EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^7])^3 - 6q*(QPochhammer[q] *QPochhammer[q^7])^3 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from first formula *)

PARI

{a(n) = local(A, t1, t2, t3); if( n<1, n==0, A = x * O(x^n); t1 = x * (eta(x + A) * eta(x^7 + A))^3; t2 = sum(k=1, (sqrtint(4*n + 1)  + 1)\2, 2 * x^(k*k - k), A); t3 = sum(k=1, sqrtint(n), 2 * x^(k*k), 1 + A); A = x * O(x^(n\7)); polcoeff( (t3 * subst(t3 + A, x, x^7) + x^2 * t2 * subst(t2 + A, x, x^7))^3 - 6*t1, n))}; /* Michael Somos, Jun 03 2005 */

Sage

A = ModularForms( Gamma1(7), 3, prec=25) . basis(); (-21*A[0] + 4*A[1] + 21*A[2] + 105*A[3] + 224*A[4] + 441*A[5] + 672*A[6])/4 # Michael Somos, May 25 2014

Formula

a(n) = A002653(n) - 6*A002656(n).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogeneous degree 6 polynomial with 28 terms. - Michael Somos, Jun 03 2005