A133721 Triangle read by rows: T(m,n) = number of n-balanced and minimal labeled covers of a finite set of m unlabeled elements (m >= 1, 1 <= n <= m).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 1, 13, 1, 1, 1, 1, 1, 1, 1, 25, 7, 1, 1, 1, 1, 1, 1, 1, 15, 6, 3, 22, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 21, 65, 81, 7, 34, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10

Offset: 1

Author

N. J. A. Sloane, Dec 30 2007

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1
1 1
1 1 1
1 1 1 1
1 3 1 1 1
1 1 1 1 1 1
1 6 7 1 1 1 1
1 1 3 1 1 1 1 1
1 10 1 13 1 1 1 1 1
1 1 25 7 1 1 1 1 1 1
1 15 6 3 22 1 1 1 1 1 1
		

Links

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

Crossrefs

Cf. A133709. Column n=2 is essentially A000217. Columns 3, 4, 5, 6 give A133722, A133723, A133724, A133733.

Programs

Maple

A133721 := proc(m,n)
        l := ceil(m/n) ;
        c := n*ceil(m/n)-m ;
        A133713(l,c) ;
end proc: # R. J. Mathar, Nov 23 2011

Mathematica

A133713[l_, cl_] := Module[{g, k, s}, g = 1; For[k = 1, k <= cl+1, k++, s = Sum[Binomial[Binomial[l, k+1] + i-1, i]*t^(i*k), {i, 0, Ceiling[cl/k]}]; g = g*s]; g = Expand[g]; SeriesCoefficient[g, {t, 0, cl}]]; A133713[, 0] = 1; a[m, n_] := A133713[Ceiling[m/n], n*Ceiling[m/n] - m]; Table[a[m, n], {m, 1, 14}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 20 2014, after R. J. Mathar *)