A004675 Theta series of extremal even unimodular lattice in dimension 72.

1, 0, 0, 0, 6218175600, 15281788354560, 9026867482214400, 1989179450818560000, 213006159759990870000, 13144087517631410995200, 525100718690287495741440, 14756609779472604266496000, 310160311536865273422120000

Offset

0,5

Comments

The construction of such a lattice was announced by G. Nebe, Aug 12 2010. - N. J. A. Sloane, Aug 13 2010

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..1000

J.-C. Belfiore and P. Sole, A Type II lattice of norm 8 in dimension 72, arXiv:1010.4484 [cs.IT], 2010. - N. J. A. Sloane, Oct 23 2010

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

G. Nebe, An extremal even unimodular lattice of dimension 72, Preprint, arXiv:1008.2862 [math.NT], Aug 12 2010. - N. J. A. Sloane, Aug 13 2010

Example

Theta series begins 1 + 6218175600*q^8 + 15281788354560*q^10 + 9026867482214400*q^12 + 1989179450818560000*q^14 + 213006159759990870000*q^16 + 13144087517631410995200*q^18 + 525100718690287495741440*q^20 + 14756609779472604266496000*q^22 + ...

Maple

# get th2, th3, th4 = Jacobi theta constants out to degree maxd
maxd:=2001:
temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a, q, maxd):
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a, q, maxd):
th4:=series(subs(q=-q, th3), q, maxd):
# get Leech etc
t1:=th2^8+th3^8+th4^8: e8:=series(t1/2, q, maxd):
t1:=th2^8*th3^8*th4^8: delta24:=series(t1/256, q, maxd):
leech:=series(e8^3-720*delta24, q, maxd):
u1:=series(leech^3, q, maxd):
#u2:=series(leech^2*delta24, q, maxd):
u3:=series(leech*delta24^2, q, maxd):
u4:=series(delta24^3, q, maxd):
u5:=series(u1-589680*u3-78624000*u4, q, maxd);

Mathematica

terms = 13;
maxd = 2*terms;
th1 = EllipticTheta[1, 0, q];
th2 = EllipticTheta[2, 0, q];
th3 = EllipticTheta[3, 0, q];
th4 = th3 /. q -> -q;
t1 = th2^8 + th3^8 + th4^8;
e8 = Series[t1/2, {q, 0, maxd}];
t1 = th2^8*th3^8*th4^8;
delta24 = Series[t1/256, {q, 0, maxd}];
leech = Series[e8^3 - 720*delta24, {q, 0, maxd}];
u1 = Series[leech^3, {q, 0, maxd}];
u3 = Series[leech*delta24^2, {q, 0, maxd}];
u4 = Series[delta24^3, {q, 0, maxd}];
u5 = Series[u1 - 589680*u3 - 78624000*u4, {q, 0, maxd}];
CoefficientList[u5, q^2][[1 ;; terms]](* Jean-François Alcover, Jul 08 2017, adapted from Maple *)

Crossrefs

Cf. A018236.

Sequence in context: A290502 A172663 A210727 * A011524 A172534 A198807

Adjacent sequences: A004672 A004673 A004674 * A004676 A004677 A004678

Keyword

nonn

Author

N. J. A. Sloane

Status

approved