Search: id:A001205
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%I A001205 M2937 N1181
%S A001205 1,0,0,1,3,12,70,465,3507,30016,286884,3026655,34944085,438263364,
%T A001205 5933502822,86248951243,1339751921865,22148051088480,388246725873208,
%U A001205 7193423109763089,140462355821628771,2883013994348484940
%N A001205 Number of clouds with n points; number of undirected 2-regular labeled 
               graphs; or number of n X n symmetric matrices with (0,1) entries, 
               trace 0 and all row sums 2.
%C A001205 a(n) ~ n!*exp(-3/4)/sqrt(Pi*n).
%C A001205 a(n) is the number of ways of covering K_n with cycles of length >= 3. 
               Also number of 'frames' on n lines: given n lines in general position 
               (none parallel and no three concurrent), a frame is a subset of n 
               of the e C(n,2) points of intersection such that no three points 
               are on the same line. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), 
               Jul 06 2006
%D A001205 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 276 and 279.
%D A001205 Editorial note: Robinson's constant, Amer. Math. Monthly, 59 (1952), 
               296-297.
%D A001205 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.
%D A001205 Ph. Flajolet, Singular combinatorics, pp. 561-571, Proc. Internat. Congr. 
               Math., Beijing 2002, Higher Education Press, Beijing, 2002, Vol III.
%D A001205 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley 
               and Sons, N.Y., 1983, ex. 3.3.6b, 3.3.34.
%D A001205 R. Robinson, A new absolute geometric constant?, Amer. Math. Monthly, 
               58 (1951), 462-469.
%D A001205 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001205 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001205 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Example 5.2.8. Also problems 5.23 and 5.15(a), case k=3.
%D A001205 Z. Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted 
               [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]
%D A001205 H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 77, 
               Eq. 3.9.1.
%D A001205 W. A. Whitworth, Choice and Chance, Bell, 1901, p. 269, ex. 160.
%H A001205 T. D. Noe, <a href="b001205.txt">Table of n, a(n) for n=0..100</a>
%H A001205 Ph. Flajolet, <a href="http://arXiv.org/abs/math.CO/0304465">Singular 
               combinatorics</a>
%H A001205 Ph. Flajolet and A. Odlyzko, <a href="http://algo.inria.fr/flajolet/Publications/
               publist.html">Singularity analysis of generating functions</a>
%H A001205 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</
               a>, 2nd edn., Academic Press, NY, 1994, p. 86, Eq. 3.9.1.
%F A001205 E.g.f.: exp(-x/2-x^2/4)/sqrt(1-x). a(n+1)=n(a(n)+a(n-2)(n-1)/2).
%F A001205 $\frac{1}{4^{n}}\sum_{\beta =0}^{\lfloor \frac{n}{2}\rfloor }\sum_{\gamma 
               =0}^{n-2\beta }\frac{(-1)^{\beta +\gamma }2^{2\beta +\gamma }n!(2n-4\beta 
               -2\gamma )!}{\beta !\gamma !((n-2\beta -\gamma )!)^{2}}$ [From Shanzhen 
               Gao (sgao2(AT)fau.edu), Jun 05 2009]
%o A001205 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(-x/2-x^2/4+x*O(x^n))/sqrt(1-x+x*O(x^n)),
               n))
%Y A001205 Cf. A000985, A000986, A002137. A diagonal of A059441.
%Y A001205 Sequence in context: A113341 A125862 A077460 this_sequence A112320 A103366 
               A020530
%Y A001205 Adjacent sequences: A001202 A001203 A001204 this_sequence A001206 A001207 
               A001208
%K A001205 nonn,easy,nice
%O A001205 0,5
%A A001205 N. J. A. Sloane (njas(AT)research.att.com).

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