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.

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, 1568, 8, 0, 0, 0, 0, 3780, 24570, 4284, 9, 0, 0, 0, 0, 945, 69300, 142380, 11070, 10, 0, 0, 0, 0, 0, 51975, 866250, 713790, 27555, 11, 0, 0, 0, 0, 0, 10395, 1455300, 8399160, 3250500, 66792, 12

Offset: 2

Author

Geoffrey Critzer, Apr 25 2017

Keywords

• nonn
• tabl

Comments

A minimal edge cover is an edge cover such that the removal of any edge in the cover destroys the covering property.

Examples

			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,  1568,      8;
0, 0,  0,   0, 3780, 24570,   4284,     9;
0, 0,  0,   0,  945, 69300, 142380, 11070, 10;
		

Links

• Eric Weisstein's World of Mathematics, Complete Graph
• Eric Weisstein's World of Mathematics, Edge Cover
• Eric Weisstein's World of Mathematics, Edge Cover Polynomial

Crossrefs

Row sums give A053530.

First positive term in each even row is A001147.

First positive term in each odd row is A200142.

Programs

Mathematica

nn = 12; list = Range[0, nn]! CoefficientList[Series[Exp[z (Exp[x] - x - 1)], {x, 0, nn}], x];Table[Map[Drop[#, 1] &,
    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

Formula

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.