A133709 Triangle read by rows: T(m,l) = number of labeled covers of size l of a finite set of m unlabeled elements (m >= 1, 1 <= l <= 2^m - 1).

1, 1, 3, 3, 1, 7, 35, 140, 420, 840, 840, 1, 12, 131, 1435, 15225, 150570, 1351770, 10810800, 75675600, 454053600, 2270268000, 9081072000, 27243216000, 54486432000, 54486432000, 1, 18, 347, 7693, 185031, 4568046, 111793710, 2661422400

Offset: 1

Author

N. J. A. Sloane, Dec 30 2007

Keywords

• nonn
• tabf

Examples

			Triangle begins:
1
1 3 3
1 7 35 140 420 840 840
1 12 131 1435 15225 150570 1351770
		

Links

• A. P. Burger and J. H. van Vuuren, Balanced minimal covers of a finite set, Discr. Math. 307 (2007), 2853-2860.

Crossrefs

Columns are given by A055998, A133710, A133711, A133712.

Programs

Maple

A133709 := proc(m,l)
        option remember;
        if l = 1 then
                1;
        else
                add((-1)^i*binomial(l,i)*binomial(2^(l-i)+m-2,m),i=0..l-1)
                - add(combinat[stirling2](l,i)*procname(m,i),i=1..l-1) ;
        end if;
end proc:
seq(seq(A133709(m,l),l=1..2^m-1),m=1..5) ; # R. J. Mathar, Nov 23 2011

Mathematica

T[m_, l_] := T[m, l] = If[l == 1, 1, Sum[(-1)^i Binomial[l, i] Binomial[ 2^(l-i)+m-2, m], {i, 0, l-1}] - Sum[StirlingS2[l, i] T[m, i], {i, 1, l-1} ] ];
Table[T[m, l], {m, 1, 5}, {l, 1, 2^m-1}] // Flatten (* Jean-François Alcover, Apr 01 2020, from Maple *)

Formula

Burger and van Vuuren give an explicit formula.