A001205 - OEIS

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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.
(Formerly M2937 N1181)

1, 0, 0, 1, 3, 12, 70, 465, 3507, 30016, 286884, 3026655, 34944085, 438263364, 5933502822, 86248951243, 1339751921865, 22148051088480, 388246725873208, 7193423109763089, 140462355821628771, 2883013994348484940 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,5`
	COMMENTS	`a(n) ~ n!exp(-3/4)/sqrt(Pin).` `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, Jul 06 2006`
	REFERENCES	`L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 276 and 279.` `Editorial note: Robinson's constant, Amer. Math. Monthly, 59 (1952), 296-297.` `S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.` `Ph. Flajolet, Singular combinatorics, pp. 561-571, Proc. Internat. Congr. Math., Beijing 2002, Higher Education Press, Beijing, 2002, Vol III.` `I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.6b, 3.3.34.` `R. Robinson, A new absolute geometric constant?, Amer. Math. Monthly, 58 (1951), 462-469.` `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).` `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.` `Z. Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From Shanzhen Gao, Jun 05 2009]` `H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 77, Eq. 3.9.1.` `W. A. Whitworth, Choice and Chance, Bell, 1901, p. 269, ex. 160.`
	LINKS	`T. D. Noe, Table of n, a(n) for n=0..100` `Ph. Flajolet, Singular combinatorics` `Ph. Flajolet and A. Odlyzko, Singularity analysis of generating functions` `H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 86, Eq. 3.9.1.`
	FORMULA	`E.g.f.: exp(-x/2-x^2/4)/sqrt(1-x).` `a(n+1) = n(a(n)+a(n-2)(n-1)/2).` `1/4^n * sum_{b =0..floor(n/2)} sum_{g =0..n-2b} \frac{(-1)^{b+g} 2^{2b +g }n!(2n-4b -2g )!}{b! g!( (n-2*b-g)!)^2} [From Shanzhen Gao, Jun 05 2009]`
	MATHEMATICA	`m = 21; CoefficientList[ Series[ Exp[-x/2 - x^2/4] / Sqrt[1-x], {x, 0, m}], x]Table[n!, {n, 0, m}] ( From Jean-François Alcover, Jun 21 2011, after e.g.f. *)`
	PROG	`(PARI) a(n)=if(n<0, 0, n!polcoeff(exp(-x/2-x^2/4+xO(x^n))/sqrt(1-x+x*O(x^n)), n))`
	CROSSREFS	`Cf. A000985, A000986, A002137. A diagonal of A059441 and A144163.` `Sequence in context: A113341 A125862 A077460 * A112320 A103366 A175836` `Adjacent sequences: A001202 A001203 A001204 * A001206 A001207 A001208`
	KEYWORD	`nonn,easy,nice`
	AUTHOR	`N. J. A. Sloane.`
	STATUS	`approved`

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Last modified May 19 11:46 EDT 2013. Contains 225429 sequences.