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%I M3606 #46 Sep 08 2022 08:44:34
%S 1,4,24,240,4320,146880,9694080,1260230400,325139443200,
%T 167121673804800,171466837323724800,351507016513635840000,
%U 1440475753672879672320000,11803258325595576034990080000,193408190923209108909347450880000
%N a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 151.
%H Robert L. Griess Jr., <a href="http://arXiv.org/abs/math/0403480">Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices</a>, arXiv:math/0403480 [math.GR], Mar 28 2004. See Proposition 8.9.
%H J. Leech, <a href="http://dx.doi.org/10.4153/CJM-1964-065-1">Some sphere packings in higher space</a>, Canad. J. Math., 16 (1964), 657-682.
%H J. Leech & N. J. A. Sloane, <a href="/A006088/a006088.pdf">Correspondence, 1975</a>
%H C. Musès, <a href="http://dx.doi.org/10.1016/S0096-3003(97)00004-0">The dimensional family approach in (hyper)sphere packing...</a>, Applied Math. Computation 88 (1997), pp. 1-26, see p. 22.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/kiss.html">Table of highest kissing numbers known</a>
%H <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>
%F a(n) = (2+2)(2+4)(2+8)(2+16)...(2+2^n).
%F From _Paul D. Hanna_, Sep 16 2009: (Start)
%F G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1-2^k*x)];
%F contrast with:
%F 1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1+2^k*x)]. (End)
%F a(n) ~ c * 2^(n*(n+1)/2), where c = A081845. - _Vaclav Kotesovec_, Dec 31 2015
%p 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); # _Emeric Deutsch_, Dec 10 2004
%t RecurrenceTable[{a[0]==1, a[n]==(2^n + 2) a[n-1]}, a[n], {n, 0, 25}] (* _Vincenzo Librandi_, Dec 31 2015 *)
%o (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)} \\ _Paul D. Hanna_, Sep 16 2009
%o (PARI) a(n) = prod(k=1, n, 2+2^k); \\ _Michel Marcus_, Jan 01 2016
%o (Magma) I:=[4]; [1] cat [n le 1 select I[n] else (2^n + 2)*Self(n-1): n in [1..20]]; // _Vincenzo Librandi_, Dec 31 2015
%Y Cf. A028362. - _Paul D. Hanna_, Sep 16 2009
%Y Cf. A081845.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, John Leech
%E More terms from _Emeric Deutsch_, Dec 10 2004
%E Replaced arXiv URL with non-cached version - _R. J. Mathar_, Oct 23 2009
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