OFFSET
0,5
COMMENTS
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
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 410-411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 276 and 279.
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.
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Z. Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From Shanzhen Gao, Jun 05 2009] [apparently unpublished as of 2016]
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
Editorial note, Robinson's constant, Amer. Math. Monthly, 59 (1952), 296-297.
Ph. Flajolet, Singular combinatorics, arXiv:math/0304465 [math.CO], 2003.
Ph. Flajolet and A. Odlyzko, Singularity analysis of generating functions, [Research Report] RR-0826, INRIA. 1988.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
H. Richter, Analyzing coevolutionary games with dynamic fitness landscapes, arXiv preprint arXiv:1603.06374 [q-bio.PE], 2016.
R. Robinson, A new absolute geometric constant?, Amer. Math. Monthly, 58 (1951), 462-469.
Weiping Wang, Tianming Wang, An automatic approach to the generating functions for some special sequences, Ars. Comb. 116 (2018) 263, Example 4.2
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 86, Eq. 3.9.1.
FORMULA
a(n) ~ n!*exp(-3/4)/sqrt(Pi*n).
E.g.f.: exp(-x/2-x^2/4)/sqrt(1-x).
D-finite with recurrence 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-2*b} (-1)^(b+g) * 2^(2b+g) * n! * (2n-4b-2g)! / (b! * g! * (n-2b-g)!^2). - Shanzhen Gao, Jun 05 2009
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
MAPLE
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
MATHEMATICA
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. *)
PROG
(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))
(Maxima)
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);
makelist(a(n), n, 0, 12); /* Emanuele Munarini, Aug 25 2017 */
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved