A002706 - OEIS

%I #33 Sep 08 2022 08:44:31

%S 1,0,42,56,84,168,280,336,462,336,840,672,1176,1176,1386,1008,1848,

%T 2016,2058,2520,3528,2408,3108,2688,4760,3024,5880,4592,6468,4704,

%U 5040,6720,6930,6832,10080,7224,7812,7392,12600,7056,14280,11760,12040,9408

%N 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).

%C In Elkies 1999 the g.f. is denoted by theta_L. - _Michael Somos_, Nov 09 2014

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Intro. to 3rd ed.

%D 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.

%H G. C. Greubel, <a href="/A002706/b002706.txt">Table of n, a(n) for n = 0..2500</a>

%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1090/S0002-9947-1983-0716832-0">Complex and integral laminated lattices</a>, Trans. Amer. Math. Soc., 280 (1983), 463-490.

%H N. Elkies, <a href="http://www.msri.org/publications/books/Book35/files/elkies.pdf">The Klein quartic in number theory</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/P6.5.html">Home page for this lattice</a>

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

%F 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

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

%t 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 *)

%o (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 */

%o (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

%o (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 */

%Y Cf. A002653, A002656.

%K nonn

%O 0,3

%A _N. J. A. Sloane_