|
%I #15 Jan 13 2020 13:55:56
%S 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,
%T 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,
%U 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
%N 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).
%H A. P. Burger and J. H. van Vuuren, <a href="https://doi.org/10.1016/j.disc.2007.03.004">Balanced minimal covers of a finite set</a>, Discr. Math. 307 (2007), 2853-2860.
%e Triangle begins:
%e 1
%e 1 1
%e 1 1 1
%e 1 1 1 1
%e 1 3 1 1 1
%e 1 1 1 1 1 1
%e 1 6 7 1 1 1 1
%e 1 1 3 1 1 1 1 1
%e 1 10 1 13 1 1 1 1 1
%e 1 1 25 7 1 1 1 1 1 1
%e 1 15 6 3 22 1 1 1 1 1 1
%p A133721 := proc(m,n)
%p l := ceil(m/n) ;
%p c := n*ceil(m/n)-m ;
%p A133713(l,c) ;
%p end proc: # _R. J. Mathar_, Nov 23 2011
%t 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_ *)
%Y Cf. A133709. Column n=2 is essentially A000217. Columns 3, 4, 5, 6 give A133722, A133723, A133724, A133733.
%K nonn,tabl
%O 1,12
%A _N. J. A. Sloane_, Dec 30 2007
|
