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
Keywords
Examples
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 + ...
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
Crossrefs
Cf. A018236.
Programs
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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);
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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 *)
Comments