|
%I #21 Jul 10 2022 16:39:34
%S 1,0,0,4096,98256,1130496,8384512,45785088,199066704,726630400,
%T 2314125312,6606336000,17213014208,41497214976,93722075136,
%U 200284889088,407539456080,793560711168,1487412428800,2697313996800,4744432596576,8110668185600,13523169792000
%N Theta series of odd Leech lattice (the unique unimodular 24-dimensional lattice with minimal norm 3).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
%H G. C. Greubel, <a href="/A027859/b027859.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/oddLeech.html">Home page for lattice</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F theta_3(z)^24 - 48*theta_3(z)^16*Del8(z) + 48*theta_3(z)^8*Del8(z)^2 where Del8(z) = [(theta_2(z)*theta_4(z))^4]/16 is the unique cusp form of weight 8 for G(2)... SPLAG p. 187.
%F Expansion of u * (u^2 - 48*u*v + 48*v^2) in powers of q where u = phi(q)^8, v = q * psi(-q)^8 and phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Feb 01 2017
%F Expansion of 16*q * f(q)^8 * (u^4 + u^3*v + 3/16*u^2*v^2 + u*v^3 + v^4) / (u * v) in powers of q where u = f(-q)^8, v = 16*q * f(-q^4)^8 and f() is a Ramanujan theta function. - _Michael Somos_, Feb 01 2017
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 4096 (t/i)^12 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Feb 01 2017
%e G.f. = 1 + 4096*q^3 + 98256*q^4 + 1130496*q^5 + 8384512*q^6 + ...
%t a[ n_] := With[ {u = EllipticTheta[ 3, 0, q]^8, v = EllipticTheta[ 2, Pi/4, q^(1/2)]^8}, SeriesCoefficient[ u (u^2 - 3 u v + 3/16 v^2), {q, 0, n}]]; (* _Michael Somos_, Feb 01 2017 *)
%o (PARI) {a(n) = if( n<0, 0, my(A, u, v); A = x * O(x^n); u = eta(x^2 + A)^40 / (eta(x + A) * eta(x^4 + A))^16; v = x * (eta(x + A) * eta(x^4 + A))^8 / eta(x^2 + A)^8; polcoeff( u * (u^2 - 48*u*v + 48*v^2), n))}; /* _Michael Somos_, Feb 01 2017 */
%o (PARI) {a(n) = if( n<0, 0, my(A, u, v); A = x * O(x^n); u = eta(x + A)^8; v = 16*x * eta(x^4 + A)^8; polcoeff( 16*x * eta(-x + A)^8 * (u^4 + u^3*v + 3/16*u^2*v^2 + u*v^3 + v^4) / (u * v), n))}; /* _Michael Somos_, Feb 01 2017 */
%o (Magma) A := Basis( ModularForms( Gamma0(4), 12), 23); A[1] + 4096*A[4] + 98256*A[5] + 1130496*A[6] + 8384512*A[7]; /* _Michael Somos_, Feb 01 2017 */
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 16 2000
|
