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

2,2

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.]

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

Example

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;
...

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 *)

Prog

(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

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.

Sequence in context: A147824 A019081 A219454 * A241250 A097335 A255297

Adjacent sequences: A059440 A059441 A059442 * A059444 A059445 A059446

Keyword

tabf,nonn,nice

Author

N. J. A. Sloane, Feb 01 2001

Extensions

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

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

Status

approved