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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

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-Fran├žois 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

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)