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A002718 Number of bicoverings of an n-set.
(Formerly M4559 N1941)
38
1, 8, 80, 1088, 19232, 424400, 11361786, 361058000, 13386003873, 570886397340, 27681861184474, 1511143062540976, 92091641176725504, 6219762391554815200, 462595509951068027741, 37676170944802047077248 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Another description: number of proper 2-covers of [1,...,n].

REFERENCES

Cameron, Peter; Prellberg, Thomas; and Stark, Dudley; Asymptotic enumeration of 2-covers and line graphs. Discrete Math. 310 (2010), no. 2, 230-240 (see t_n).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

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).

LINKS

Table of n, a(n) for n=2..17.

L. Comtet, Birecouvrements et birevêtements d’un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]

FORMULA

E.g.f. for k-block bicoverings of an n-set is exp(-x-1/2*x^2*(exp(y)-1))*Sum_{i=0..inf} x^i/i!*exp(binomial(i, 2)*y).

Stirling_2 transform of A060053.

The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(e^x-1).

MATHEMATICA

nmax = 16; imax = 2*(nmax - 2); egf := E^(-x - 1/2*x^2*(E^y - 1))*Sum[(x^i/i!)*E^(Binomial[i, 2]*y), {i, 0, imax}]; fx = CoefficientList[Series[egf, {y, 0, imax}], y]*Range[0, imax]!; a[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3] // Total; Table[ a[n] , {n, 2, nmax}] (* Jean-François Alcover, Apr 04 2013 *)

CROSSREFS

Cf. A020554, A002719, A003462, A059945-A059951, A060053. Row sums of A059443.

Sequence in context: A234596 A060375 A097815 * A222825 A057707 A222671

Adjacent sequences:  A002715 A002716 A002717 * A002719 A002720 A002721

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vladeta Jovovic, Feb 18 2001

STATUS

approved

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Last modified July 27 03:28 EDT 2016. Contains 275064 sequences.