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A281269 Triangle read by rows: T(n,k) is the number of edge covers of the complete labeled graph on n nodes that are minimal and have exactly k edges, n>=2, 1<=k<=n-1.

%I A281269 #30 Feb 16 2025 08:33:39
%S A281269 1,0,3,0,3,4,0,0,30,5,0,0,15,150,6,0,0,0,315,525,7,0,0,0,105,3360,
%T A281269 1568,8,0,0,0,0,3780,24570,4284,9,0,0,0,0,945,69300,142380,11070,10,0,
%U A281269 0,0,0,0,51975,866250,713790,27555,11,0,0,0,0,0,10395,1455300,8399160,3250500,66792,12
%N A281269 Triangle read by rows: T(n,k) is the number of edge covers of the complete labeled graph on n nodes that are minimal and have exactly k edges, n>=2, 1<=k<=n-1.
%C A281269 A minimal edge cover is an edge cover such that the removal of any edge in the cover destroys the covering property.
%H A281269 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H A281269 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCover.html">Edge Cover</a>
%H A281269 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCoverPolynomial.html">Edge Cover Polynomial</a>
%F A281269 E.g.f.: exp(y*x^2/2) * Sum_{j>=0} (y*x)^j/j! * Sum_{k=0..floor(j/2)} A008299(j,k)*x^k.
%e A281269 1;
%e A281269 0, 3;
%e A281269 0, 3,  4;
%e A281269 0, 0, 30,   5;
%e A281269 0, 0, 15, 150,    6;
%e A281269 0, 0,  0, 315,  525,     7;
%e A281269 0, 0,  0, 105, 3360,  1568,      8;
%e A281269 0, 0,  0,   0, 3780, 24570,   4284,     9;
%e A281269 0, 0,  0,   0,  945, 69300, 142380, 11070, 10;
%t A281269 nn = 12; list = Range[0, nn]! CoefficientList[Series[Exp[z (Exp[x] - x - 1)], {x, 0, nn}], x];Table[Map[Drop[#, 1] &,
%t A281269     Drop[Range[0, nn]! CoefficientList[Series[Exp[u z^2/2!] Sum[(u z)^j/j!*list[[j + 1]], {j, 0, nn}], {z, 0, nn}], {z, u}], 2]][[n, 1 ;; n]], {n, 1, nn - 1}] // Grid
%Y A281269 Row sums give A053530.
%Y A281269 First positive term in each even row is A001147.
%Y A281269 First positive term in each odd row is A200142.
%K A281269 nonn,tabl
%O A281269 2,3
%A A281269 _Geoffrey Critzer_, Apr 25 2017