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A004675 Theta series of extremal even unimodular lattice in dimension 72. 4
1, 0, 0, 0, 6218175600, 15281788354560, 9026867482214400, 1989179450818560000, 213006159759990870000, 13144087517631410995200, 525100718690287495741440, 14756609779472604266496000, 310160311536865273422120000 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 22 19:48 EDT 2024. Contains 372758 sequences. (Running on oeis4.)