%I M1903
%S 2,8,96,43008,58018928640,4587291356489073135452160
%N Number of directed Hamiltonian cycles (or Gray codes) on ncube with a marked starting node.
%C More precisely, this is the number of ways of making a list of the 2^n nodes of the ncube, with a distinguished starting position and a direction, such that each node is adjacent to the previous one and the last node is adjacent to the first.
%D M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 24.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H H. Haanpaa and Patric R. J. Östergård, Counting Hamiltonian cycles in bipartite graphs, Math. Comp. 83 (2014), 979995. doi:<a href="http://dx.doi.org/10.1090/S00255718201302741X">S00255718201302741X</a>
%H D. Sensarma, S. S. Sarma, <a href="http://ijret.org/Volumes/V03/I03/IJRET_110303121.pdf">GMDES: A graph based modified Data Encryption Standard algorithm with enhanced security</a>, IJRET: International Journal of Research in Engineering and Technology 03:03 (2014), 653660. See Section 2.2.
%H Michel Deza and Roman Shklyar, <a href="http://arxiv.org/abs/1003.4391">Enumeration of Hamiltonian Cycles in 6cube</a>, arXiv:1003.4391v1 [There may be errors  see Haanpaa and Ostergard, 2012]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>
%F a(n) = A003042(n)*2^n.  _Max Alekseyev_, Jun 15 2006
%e a(1) = 2: we have 1,2 or 2,1.
%e a(2) = 8: label the nodes 1, 2, ..., 4. Then the 8 possibilities are 1,2,3,4; 1,4,3,2; 2,3,4,1; 2,1,4,3; etc.
%Y Cf. A003042, A006070, A091299, A091302, A159344.
%K nonn,more
%O 1,1
%A _N. J. A. Sloane_.
%E a(5) corrected by Jonathan Cross (jcross(AT)wcox.com), Oct 10 2001
%E Definition corrected by _Max Alekseyev_, Jun 15 2006
%E a(6) from Michel Deza, Mar 28 2010
%E a(6) corrected by Haanpaa and Östergård, 2012.  _N. J. A. Sloane_, Sep 06 2012
