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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) 20

%I M2937 N1181

%S 1,0,0,1,3,12,70,465,3507,30016,286884,3026655,34944085,438263364,

%T 5933502822,86248951243,1339751921865,22148051088480,388246725873208,

%U 7193423109763089,140462355821628771,2883013994348484940

%N 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 a(n) ~ n!*exp(-3/4)/sqrt(Pi*n).

%C 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

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 410-411.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 276 and 279.

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.

%D Ph. Flajolet, Singular combinatorics, pp. 561-571, Proc. Internat. Congr. Math., Beijing 2002, Higher Education Press, Beijing, 2002, Vol III.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.6b, 3.3.34.

%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).

%D 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%D Z. Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From _Shanzhen Gao_, Jun 05 2009] [apparently unpublished as of 2016]

%D H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 77, Eq. 3.9.1.

%D W. A. Whitworth, Choice and Chance, Bell, 1901, p. 269, ex. 160.

%H T. D. Noe, <a href="/A001205/b001205.txt">Table of n, a(n) for n=0..100</a>

%H Editorial note, <a href="http://www.jstor.org/stable/2307503">Robinson's constant</a>, Amer. Math. Monthly, 59 (1952), 296-297.

%H Ph. Flajolet, <a href="https://arxiv.org/abs/math/0304465">Singular combinatorics</a>, arXiv:math/0304465 [math.CO], 2003.

%H Ph. Flajolet and A. Odlyzko, <a href="https://hal.inria.fr/inria-00075725">Singularity analysis of generating functions</a>, [Research Report] RR-0826, INRIA. 1988.

%H Rui-Li Liu, Feng-Zhen Zhao, <a href="https://www.emis.de/journals/JIS/VOL21/Liu/liu19.html">New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.

%H H. Richter, <a href="https://arxiv.org/abs/1603.06374">Analyzing coevolutionary games with dynamic fitness landscapes</a>, arXiv preprint arXiv:1603.06374 [q-bio.PE], 2016.

%H R. Robinson, <a href="http://www.jstor.org/stable/2306924">A new absolute geometric constant?</a>, Amer. Math. Monthly, 58 (1951), 462-469.

%H 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 E.g.f.: exp(-x/2-x^2/4)/sqrt(1-x).

%F D-finite with recurrence a(n+1) = n*(a(n)+a(n-2)*(n-1)/2).

%F 1/4^n * Sum_{b=0..floor(n/2)} Sum_{g=0..n-2*b} (-1)^(b+g) * 2^(2b+g) * n! * (2n-4b-2g)! / (b! * g! * (n-2b-g)!^2). - _Shanzhen Gao_, Jun 05 2009

%F a(n) = (-1)^n*n!*Sum_{k=0..n}(3/4)^k*binomial(-1/2, n - k)*hypergeom([1/2, -k], [1/2 - n + k], 1/3)/ k!. - _Peter Luschny_, Aug 26 2017

%p a := n -> (-1)^n*n!*add((3/4)^k*binomial(-1/2, n-k)*hypergeom([1/2,-k], [1/2-n+k], 1/3)/ k!, k=0..n): seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Aug 26 2017

%t m = 21; CoefficientList[ Series[ Exp[-x/2 - x^2/4] / Sqrt[1-x], {x, 0, m}], x]*Table[n!, {n, 0, m}] (* _Jean-François Alcover_, Jun 21 2011, after e.g.f. *)

%o (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))

%o (Maxima)

%o a(n):=sum(sum(binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-i),i,0,k),k,0,n);

%o makelist(a(n),n,0,12); /* _Emanuele Munarini_, Aug 25 2017 */

%Y Cf. A000985, A000986, A002137. A diagonal of A059441 and A144163.

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_

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Last modified June 4 03:40 EDT 2020. Contains 334815 sequences. (Running on oeis4.)