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A034597
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Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.
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5
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1, 196560, 52416000, 6218175600, 565866362880, 45792819072000, 3486157968384000, 256206274225902000, 18422726047165440000, 1305984407917646096640, 91692325887531393024000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane, "Upper bounds for modular forms, lattices and codes", J. Alg., 36 (1975), 68-76.
C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..100
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
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EXAMPLE
| When n=1 we get the theta series of the 24-dimensional Leech lattice: 1+196560*q^4+16773120*q^6+... (see A008408). For n=2 we get A004672 and for n=3, A004675.
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MAPLE
| # Extremal theta series:
read(`/usr/njas/bin/format`); kernelopts(printbytes=false): interface(screenwidth=200); with(numtheory);
# set mu:
for mu from 1 to 100 do
# set max deg:
md:=mu+3;
f:=1+240*add(sigma[3](i)*x^i, i=1..md);
f:=series(f, x, md);
f:=series(f^3, x, md);
g:=series(x*mul( (1-x^i)^24, i=1..md), x, md);
W0:=series(f^mu, x, md):
h:=series(g/f, x, md):
A:=series(W0, x, md):
Z:=A:
for i from 1 to mu do
Z:=series(Z*h, x, md);
A:=series(A-coeff(A, x, i)*Z, x, md);
od:
lprint(A);
od:
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CROSSREFS
| Cf. A034598 (second coefficient, which eventually becomes negative), A034414, A034415.
Sequence in context: A179253 A008408 A001942 * A037148 A175744 A024211
Adjacent sequences: A034594 A034595 A034596 * A034598 A034599 A034600
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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