A004046 Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.

1, 0, 0, 26208, 530712, 6368544, 47331648, 256864608, 1116087336, 4092877152, 12996075456, 37058557536, 96952754808, 232778774592, 526258264896, 1128148021728, 2286143305992, 4451523096384, 8386247967552, 15130902687264, 26614339616592, 45684687301344

Offset

0,4

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 21 2015

References

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

G. Nebe and N. J. A. Sloane, Home page for lattice

H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory, 54 (1995), 190-202.

N. J. A. Sloane, Seven Staggering Sequences.

Formula

Theta series = a^12 - 9/2*a^8*b^4 + 414*a^6*b^6 + 1458*a^4*b^8 + 1998*a^2*b^10 + 459/2*b^12 (see PARI code for details).

G.f.: (27*a(x)^12 - 72*a(x)^9*b(x)^3 + 64*a(x)^6*b(x)^6 + 16*a(x)^3*b(x)^9 - 8*b(x)^12) / 27 where a(), b() are cubic AGM theta functions, - Michael Somos, Dec 25 2015

Example

G.f. = 1 + 26208*x^3 + 530712*x^4 + 6368544*x^5 + 47331648*x^6 + ...
G.f. = 1 + 26208*q^6 + 530712*q^8 + 6368544*q^10 + 47331648*q^12 + ...

Mathematica

a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = ( 1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^4 (27 z^4 - 72 z^3 + 64 z^2 + 16 z - 8) / 27, {q, 0, n}]]]; (* Michael Somos, Dec 25 2015 *)

Prog

(PARI) th3 = sum(n=1, noo\2, 2*x^(4*n^2), 1+A);
th4 = sum(n=1, noo\2, (-1)^n*2*x^(4*n^2), 1+A);
th2 = sum(n=0, noo\2, 2*x^(4*n^2+4*n+1), A);
chk("th3^4 == th4^4+th2^4");
/* A004016(x^4) */
phi0 = th2*subst(th2, x, x^3)+ th3*subst(th3, x, x^3);
/* 2*x*A033762(x^2) */
phi1 = th2*subst(th3, x, x^3)+ th3*subst(th2, x, x^3);
/* A004010(x^2) */
K_12 = phi0^6+45*phi0^2*phi1^4+18*phi1^6;
a=phi0; b=phi1;
A004046=a^12-9/2*a^8*b^4+414*a^6*b^6+1458*a^4*b^8+1998*a^2*b^10+459/2*b^12;
(Magma) A := Basis( ModularForms( Gamma1(3), 12), 22);  A[1] + 26208*A[4] + 530712*A[5]; /* Michael Somos, Dec 21 2015 */
(PARI) {a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^4 * (27*z^4 - 72*z^3 + 64*z^2 + 16*z - 8) / 27, n))}; /* Michael Somos, Dec 25 2015 */

Crossrefs

Cf. A107657.

Sequence in context: A235900 A245787 A034622 * A031652 A236787 A237003

Adjacent sequences: A004043 A004044 A004045 * A004047 A004048 A004049

Keyword

nonn

Author

N. J. A. Sloane

Extensions

PARI code from Michael Somos, Jun 07 2005

Status

approved