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%I M4420 N1867
%S 0,0,0,1,7,37,197,1172,8018,62814,556014,5488059,59740609,710771275,
%T 9174170011,127661752406,1904975488436,30341995265036,513771331467372,
%U 9215499383109573,174548332364311563,3481204991988351553,72920994844093191553,1600596371590399671784
%N Sum_{k=3..n} (k-1)!*C(n,k)/2.
%C Maximal number of cycles in complete graph on n nodes. - _Erich Friedman_.
%C Number of equations that must be checked to verify reversibility of an n state Markov chain using the Kolmogorov criterion [From Qian Jiang (jiang1h(AT)uwindsor.ca), Jun 08 2009]
%D J. P. Char, Master circuit matrix, Proc. IEE, 115 (1968), 762-770.
%D F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282.
%D E.P.C. Kao, An Introduction to Stochastic Processes, Duxbury Press, 1997, 209-210. [From Qian Jiang (jiang1h(AT)uwindsor.ca), Jun 08 2009]
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002807/b002807.txt">Table of n, a(n) for n=0..100</a>
%H P. Pollack, <a href="http://www.math.dartmouth.edu/~ppollack/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, ch. 7. [?Broken link]
%H P. Pollack, <a href="http://alpha01.dm.unito.it/personalpages/cerruti/ac/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, ch. 7.
%H M. Scullard, <a href="http://www.math.ucsd.edu/~williams/courses/m28908/scullardMath289_Reversibility.pdf"> Reversible Markov Chains</a> [From Qian Jiang (jiang1h(AT)uwindsor.ca), Jun 08 2009]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%F E.g.f.: (-1/4)*exp(x)*(2*ln(1-x)+2*x+x^2). - _Vladeta Jovovic_, Oct 26 2004
%F a(n)=(n-1)*(n-2)/2+n*a(n-1)-(n-1)*a(n-2). - _Vladeta Jovovic_, Jan 22 2005
%t Table[Sum[((k-1)!Binomial[n,k])/2,{k,3,n}],{n,0,25}] (* From Harvey P. Dale, June 24 2011 *)
%t a[n_] := n/4*(2*HypergeometricPFQ[{1, 1, 1-n}, {2}, -1] - n - 1); a[0]=0; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 05 2012 *)
%Y Cf. A117130, A099198, A099201, A070968.
%K nonn,easy,nice,changed
%O 0,5
%A _N. J. A. Sloane_.
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