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A002807 a(n) = Sum_{k=3..n} (k-1)!*C(n,k)/2.
(Formerly M4420 N1867)
12
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
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
Sequence in context: A069378 A287808 A117130 * A124610 A002683 A319013
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)