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A094573 Triangle T(n,k) giving number of (<=2)-covers of an n-set with k blocks.

%I A094573 #20 Jul 11 2024 10:17:24
%S A094573 1,1,1,3,1,1,12,20,7,1,39,169,186,59,3,1,120,1160,2755,2243,661,55,1,
%T A094573 363,7381,33270,52060,33604,9167,910,15,1,1092,45500,367087,988750,
%U A094573 1126874,601262,151726,16401,525,1,3279,276529,3873786,17005149
%N A094573 Triangle T(n,k) giving number of (<=2)-covers of an n-set with k blocks.
%C A094573 Cover of a set is (<=2)-cover if every element of the set is covered with at most two blocks of the cover.
%F A094573 E.g.f.: exp(-x-x^2/2*(exp(y)-1))*(Sum_{n>=0} exp(y*binomial(n+1, 2))*x^n/n!).
%e A094573 Triangle T(n,k) begins:
%e A094573   1;
%e A094573   1;
%e A094573   1,   3,    1;
%e A094573   1,  12,   20,    7;
%e A094573   1,  39,  169,  186,   59,   3;
%e A094573   1, 120, 1160, 2755, 2243, 661, 55;
%e A094573   ...
%t A094573 rows = 9; m = rows + 2;
%t A094573 egf = Exp[-x - (x^2/2)*(Exp[y]-1)]*Sum[Exp[y*Binomial[n+1, 2]]*(x^n/n!), {n, 0, m}];
%t A094573 cc = CoefficientList[# + O[x]^m, x]& /@ CoefficientList[egf + O[y]^m, y];
%t A094573 (Range[0, Length[cc]-1]! * cc)[[1 ;; rows]] /. {0, a__} :> {a} // Flatten (* _Jean-François Alcover_, May 13 2019 *)
%Y A094573 Row sums give A094574.
%Y A094573 Cf. A059443, A060052.
%K A094573 nonn,tabf
%O A094573 0,4
%A A094573 Goran Kilibarda, _Vladeta Jovovic_, May 12 2004