A059443 Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.

1, 4, 4, 13, 39, 25, 3, 40, 280, 472, 256, 40, 121, 1815, 6185, 7255, 3306, 535, 15, 364, 11284, 70700, 149660, 131876, 51640, 8456, 420, 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105, 3280, 416560, 7894992, 44659776, 103290096

Offset: 2

Author

N. J. A. Sloane, Feb 01 2001

Keywords

• tabf
• nonn
• nice

Examples

			T(2,3) = 1: 1|12|2.
T(3,3) = 4: 1|123|23, 12|13|23, 12|123|3, 123|13|2.
T(3,4) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.
Triangle T(n,k) begins:
:    1;
:    4,     4;
:   13,    39,     25,       3;
:   40,   280,    472,     256,      40;
:  121,  1815,   6185,    7255,    3306,     535,     15;
:  364, 11284,  70700,  149660,  131876,   51640,   8456,    420;
: 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105;
...
		

References

• 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.

Links

• Alois P. Heinz, Rows n = 2..60, flattened
• 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.]

Crossrefs

Columns k=3-10 give: A003462, A059945, A059946, A059947, A059948, A059949, A059950, A059951.

Row sums are A002718.

Main diagonal gives A275517.

Right border gives A275521.

Programs

Mathematica

nmax = 8; 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]!; row[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3]; Table[ row[n], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Sep 21 2012 *)

PARI

\ps 22;
s = 8; pv = vector(s); for(n=1,s,pv[n]=round(polcoeff(f(x,y),n,y)*n!));
for(n=1,s,for(m=3,poldegree(pv[n],x),print1(polcoeff(pv[n],m),", "))) \\ Gerald McGarvey, Dec 03 2009

Formula

E.g.f. for m-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).

T(n, k) = Sum{j=0..n} Stirling2(n, j) * A060052(j, k). - David Pasino, Sep 22 2016

Extensions

More terms and additional comments from Vladeta Jovovic, Feb 14 2001

a(37) corrected by Gerald McGarvey, Dec 03 2009