cp's OEIS Frontend

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).

%I A133709 #14 Apr 01 2020 14:05:29
%S A133709 1,1,3,3,1,7,35,140,420,840,840,1,12,131,1435,15225,150570,1351770,
%T A133709 10810800,75675600,454053600,2270268000,9081072000,27243216000,
%U A133709 54486432000,54486432000,1,18,347,7693,185031,4568046,111793710,2661422400
%N 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).
%H A133709 A. P. Burger and J. H. van Vuuren, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.004">Balanced minimal covers of a finite set</a>, Discr. Math. 307 (2007), 2853-2860.
%F A133709 Burger and van Vuuren give an explicit formula.
%e A133709 Triangle begins:
%e A133709 1
%e A133709 1 3 3
%e A133709 1 7 35 140 420 840 840
%e A133709 1 12 131 1435 15225 150570 1351770
%p A133709 A133709 := proc(m,l)
%p A133709         option remember;
%p A133709         if l = 1 then
%p A133709                 1;
%p A133709         else
%p A133709                 add((-1)^i*binomial(l,i)*binomial(2^(l-i)+m-2,m),i=0..l-1)
%p A133709                 - add(combinat[stirling2](l,i)*procname(m,i),i=1..l-1) ;
%p A133709         end if;
%p A133709 end proc:
%p A133709 seq(seq(A133709(m,l),l=1..2^m-1),m=1..5) ; # _R. J. Mathar_, Nov 23 2011
%t A133709 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} ] ];
%t A133709 Table[T[m, l], {m, 1, 5}, {l, 1, 2^m-1}] // Flatten (* _Jean-François Alcover_, Apr 01 2020, from Maple *)
%Y A133709 Columns are given by A055998, A133710, A133711, A133712.
%K A133709 nonn,tabf
%O A133709 1,3
%A A133709 _N. J. A. Sloane_, Dec 30 2007