%I #12 Aug 16 2024 21:11:37
%S 1,2,1,4,3,2,7,7,6,3,11,13,14,9,4,16,22,26,21,13,5,23,34,44,40,31,17,
%T 7,31,50,68,68,59,41,23,8,41,70,100,106,101,79,55,28,10,53,95,140,157,
%U 158,136,106,68,35,12,67,125,190,221,234,214,182,132,85,42,14,83,161
%N Triangle T(n,k) of number of minimal 3-covers of an unlabeled n+3-set that cover k points of that set uniquely (k=3,..,n+3).
%C Row sums give A005783.
%H V. Jovovic, <a href="/A056885/a056885.pdf">More information</a>
%F T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^3/6*(Z(S_n; 5+3*x, 5+3*x^2, ...)+3*Z(S_n; 3+x, 5+3*x^2, 3+x^3, 5+3*x^4, ...)+2*Z(S_n; 2, 2, 5+3*x^3, 2, 2, 5+3*x^6, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
%e [1], [2, 1], [4, 3, 2], [7, 7, 6, 3], ...
%e There are 7 minimal 3-covers of an unlabeled 6-set that cover 3 points of that set uniquely: {{1}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 4, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}.
%Y Cf. A001752, A056885, A057222, A057223, A057524.
%K nonn,tabl
%O 0,2
%A _Vladeta Jovovic_, Oct 16 2000