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A282575 Triangular array read by rows. T(n,k) is the number of minimal covers of an n-set with exactly k points that are in more than one set of the cover, n>=0, 0<=k<=max(0,n-2).

%I A282575 #19 Aug 08 2022 21:27:43
%S A282575 1,1,2,5,3,15,28,6,52,210,190,10,203,1506,3360,1340,15,877,10871,
%T A282575 48321,60270,9065,21,4140,80592,636300,1820056,1132880,57512,28,21147,
%U A282575 618939,8081928,45455676,76834926,21067452,344316,36,115975,4942070,101684115,1027544400,3860929170,3406410252,377190240,1966440,45
%N A282575 Triangular array read by rows.  T(n,k) is the number of minimal covers of an n-set with exactly k points that are in more than one set of the cover, n>=0, 0<=k<=max(0,n-2).
%H A282575 Alois P. Heinz, <a href="/A282575/b282575.txt">Rows n = 0..100, flattened</a>
%H A282575 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 A282575 E.g.f.: (exp(x) - 1)^n/n!*exp(y*(2^n - n - 1)*x).
%e A282575 Triangle T(n,k) begins:
%e A282575 :    1;
%e A282575 :    1;
%e A282575 :    2;
%e A282575 :    5,     3;
%e A282575 :   15,    28,      6;
%e A282575 :   52,   210,    190,      10;
%e A282575 :  203,  1506,   3360,    1340,      15;
%e A282575 :  877, 10871,  48321,   60270,    9065,    21;
%e A282575 : 4140, 80592, 636300, 1820056, 1132880, 57512, 28;
%p A282575 T:= (n, k)-> binomial(n, k)*add(Stirling2(n-k, j)*(2^j-j-1)^k, j=0..n-k):
%p A282575 seq(seq(T(n,k), k=0..max(0,n-2)), n=0..12);  # _Alois P. Heinz_, Feb 18 2017
%t A282575 nn = 8; Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! Exp[y (2^n - n - 1) x], {n, 0,nn}], {x, 0, nn}], {x, y}]], 1] // Grid
%Y A282575 Cf. A035348. Row sums A046165. Column k=0 A000110.  Column k=1 A003466.
%Y A282575 Mirrored triangle gives A035347.
%K A282575 nonn,tabf
%O A282575 0,3
%A A282575 _Geoffrey Critzer_, Feb 18 2017