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A002807 a(n) = Sum_{k=3..n} (k-1)!*C(n,k)/2.
(Formerly M4420 N1867)
10
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

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

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.

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

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: A196805 A069378 A117130 * A124610 A002683 A126475

Adjacent sequences:  A002804 A002805 A002806 * A002808 A002809 A002810

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 23 02:46 EDT 2017. Contains 286909 sequences.