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A002807
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a(n) = Sum_{k=3..n} (k-1)!*C(n,k)/2.
(Formerly M4420 N1867)
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12
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0, 0, 0, 1, 7, 37, 197, 1172, 8018, 62814, 556014, 5488059, 59740609, 710771275, 9174170011, 127661752406, 1904975488436, 30341995265036, 513771331467372, 9215499383109573, 174548332364311563, 3481204991988351553, 72920994844093191553, 1600596371590399671784
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(Magma) [&+[Factorial(k-1)*Binomial(n, k) div 2: k in [3..n]]: n in [3..30]]; // Vincenzo Librandi, Mar 06 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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