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 A002807 a(n) = Sum_{k=3..n} (k-1)!*C(n,k)/2. (Formerly M4420 N1867) 11
 0, 0, 0, 1, 7, 37, 197, 1172, 8018, 62814, 556014, 5488059, 59740609, 710771275, 9174170011, 127661752406, 1904975488436, 30341995265036, 513771331467372, 9215499383109573, 174548332364311563, 3481204991988351553, 72920994844093191553, 1600596371590399671784 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Number of cycles in the complete graph on n nodes K_{n}. - Erich Friedman Number of equations that must be checked to verify reversibility of an n state Markov chain using the Kolmogorov criterion. - Qian Jiang (jiang1h(AT)uwindsor.ca), Jun 08 2009 Also the number of paths in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017 REFERENCES E.P.C. Kao, An Introduction to Stochastic Processes, Duxbury Press, 1997, 209-210. [From Qian Jiang (jiang1h(AT)uwindsor.ca), Jun 08 2009] N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..100 P. H. Brill, Chi ho Cheung, Myron Hlynka, Q. Jiang, Reversibility Checking for Markov Chains, Communications on Stochastic Analysis (2018) Vol. 12, No. 2, Art. 2, 129-135. J. P. Char, Master circuit matrix, Proc. IEE, 115 (1968), 762-770. F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282. Q. Jiang, M. Hlynka, P.H. Brill, C.H. Cheung, Reversibility Checking for Markov Chains, arXiv:1806.10154 [math.PR], 2018. P. Pollack, Analytic and Combinatorial Number Theory Course Notes, ch. 7. [?Broken link] P. Pollack, Analytic and Combinatorial Number Theory Course Notes, ch. 7. M. Scullard, Reversible Markov Chains [From Qian Jiang (jiang1h(AT)uwindsor.ca), Jun 08 2009] Eric Weisstein's World of Mathematics, Complete Graph Eric Weisstein's World of Mathematics, Graph Cycle Eric Weisstein's World of Mathematics, Graph Path FORMULA E.g.f.: (-1/4)*exp(x)*(2*log(1-x)+2*x+x^2). - Vladeta Jovovic, Oct 26 2004 a(n) = (n-1)*(n-2)/2 + n*a(n-1) - (n-1)*a(n-2). - Vladeta Jovovic, Jan 22 2005 a(n) ~ exp(1)/2 * (n-1)! * (1 + 1/n + 2/n^2 + 5/n^3 + 15/n^4 + 52/n^5 + 203/n^6 + 877/n^7 + 4140/n^8 + 21147/n^9 + ...). Coefficients are the Bell numbers (A000110). - Vaclav Kotesovec, Mar 08 2016 For n>2, a(n) = Sum_{k=1..n-2} A000522(k-1)*A000217(k). - Vaclav Kotesovec, Mar 08 2016 MATHEMATICA Table[Sum[((k-1)!Binomial[n, k])/2, {k, 3, n}], {n, 0, 25}] (* Harvey P. Dale, Jun 24 2011 *) 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 *) PROG (MAGMA) [&+[Factorial(k-1)*Binomial(n, k) div 2: k in [3..n]]: n in [3..30]]; // Vincenzo Librandi, Mar 06 2016 (PARI) a(n)=sum(k=3, n, (k-1)!*binomial(n, k)/2) \\ Charles R Greathouse IV, Feb 08 2017 CROSSREFS Cf. A284947 (triangle of k-cycle counts in K_n). - Eric W. Weisstein, Apr 06 2017 Cf. A117130, A099198, A099201, A070968. Sequence in context: A069378 A287808 A117130 * A124610 A002683 A319013 Adjacent sequences:  A002804 A002805 A002806 * A002808 A002809 A002810 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified January 17 09:54 EST 2020. Contains 330949 sequences. (Running on oeis4.)