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A006088
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a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n).
(Formerly M3606)
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3
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1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000, 193408190923209108909347450880000
(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, p. 151.
J. Leech, Some sphere packings in higher space, Canad. J. Math., 16 (1964), 657-682.
C. Muses, The dimensional family approach in (hyper)sphere packing..., Applied Math. Computation 88 (1997), pp. 1-26, see p. 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Robert L. Griess Jr. Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices. Mar 28 2004. See Proposition 8.9.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
Index entries for sequences related to Barnes-Wall lattices
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FORMULA
| (2+2)(2+4)(2+8)(2+16)...(2+2^n ).
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2009: (Start)
G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1-2^k*x)];
contrast with:
1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1+2^k*x)]. (End)
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MAPLE
| a[0]:=1: for n from 1 to 16 do a[n]:=(2^n+2)*a[n-1] od: seq(a[n], n=0..16); (Deutsch)
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PROG
| (PARI) {a(n)=polcoeff(sum(m=0, n, 2^(m*(m+1)/2)*x^m/prod(k=1, m+1, 1-2^k*x+x*O(x^n))), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2009]
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CROSSREFS
| Cf. A028362. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2009]
Sequence in context: A052718 A061640 A126391 * A141013 A176785 A095340
Adjacent sequences: A006085 A006086 A006087 * A006089 A006090 A006091
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), John Leech
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2004
Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009
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