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A035347 Triangle of a(n,k) = number of minimal covers of an n-set that cover k points of that set uniquely (n >= 1, k >= 1).

%I A035347 #17 Mar 04 2020 22:38:49
%S A035347 1,0,2,0,3,5,0,6,28,15,0,10,190,210,52,0,15,1340,3360,1506,203,0,21,
%T A035347 9065,60270,48321,10871,877,0,28,57512,1132880,1820056,636300,80592,
%U A035347 4140,0,36,344316,21067452,76834926,45455676,8081928,618939,21147,0,45
%N A035347 Triangle of a(n,k) = number of minimal covers of an n-set that cover k points of that set uniquely (n >= 1, k >= 1).
%H A035347 T. Hearne and C. G. Wagner, <a href="http://dx.doi.org/10.1016/0012-365X(73)90141-6">Minimal covers of finite sets</a>, Discr. Math. 5 (1973), 247-251.
%F A035347 a(n, k) = C(n, k)*Sum_{j=1..k} S(k, j)*(2^j-j-1)^(n-k), where S(k, j) are Stirling numbers of the second kind.
%F A035347 E.g.f.: Sum_{k>=1} (exp(y*x) - 1)^k/k! * exp((2^k-k-1)x). - _Geoffrey Critzer_, Jun 28 2013
%e A035347 1; 0,2; 0,3,5; 0,6,28,15; ...
%t A035347 a[n_, k_] := Binomial[n, k] * Sum[ StirlingS2[k, j]*(2^j - j - 1)^(n - k), {j, 1, k}]; a[n_, n_] := Sum[ StirlingS2[n, j], {j, 1, n}]; Flatten[ Table[a[n, k], {n, 1, 10}, {k, 1, n}]] (* _Jean-François Alcover_, Jun 26 2012, from formula *)
%Y A035347 Cf. A056885 for unlabeled case. Row sums give A046165.
%K A035347 nonn,tabl,easy,nice
%O A035347 1,3
%A A035347 _N. J. A. Sloane_
%E A035347 More terms from _Vladeta Jovovic_, Sep 06 2000