login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002807 a(n) = Sum_{k=3..n} (k-1)!*C(n,k)/2.
(Formerly M4420 N1867)
11

%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 a(n) = Sum_{k=3..n} (k-1)!*C(n,k)/2.

%C Number of cycles in the complete graph on n nodes K_{n}. - _Erich Friedman_

%C 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

%C Also the number of paths in the (n-1)-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017

%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 J. P. Char, <a href="http://dx.doi.org/10.1049/piee.1968.0138">Master circuit matrix</a>, Proc. IEE, 115 (1968), 762-770.

%H F. C. Holroyd and W. J. G. Wingate, <a href="http://dx.doi.org/10.1016/S0012-365X(85)80003-0">Cycles in the complement of a tree or other graph</a>, Discrete Math., 55 (1985), 267-282.

%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>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphPath.html">Graph Path</a>

%F E.g.f.: (-1/4)*exp(x)*(2*log(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

%F 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

%F For n>2, a(n) = Sum_{k=1..n-2} A000522(k-1)*A000217(k). - _Vaclav Kotesovec_, Mar 08 2016

%t Table[Sum[((k-1)!Binomial[n,k])/2,{k,3,n}],{n,0,25}] (* _Harvey P. Dale_, Jun 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-Fran├žois Alcover_, Oct 05 2012 *)

%o (MAGMA) [&+[Factorial(k-1)*Binomial(n,k) div 2: k in [3..n]]: n in [3..30]]; // _Vincenzo Librandi_, Mar 06 2016

%o (PARI) a(n)=sum(k=3,n, (k-1)!*binomial(n,k)/2) \\ _Charles R Greathouse IV_, Feb 08 2017

%Y Cf. A284947 (triangle of k-cycle counts in K_n). - _Eric W. Weisstein_, Apr 06 2017

%Y Cf. A117130, A099198, A099201, A070968.

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 01:16 EST 2017. Contains 294837 sequences.