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

2,3

Comments

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

Links

Table of n, a(n) for n=2..67.

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

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.

Example

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;

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

Crossrefs

Row sums give A053530.

First positive term in each even row is A001147.

First positive term in each odd row is A200142.

Sequence in context: A278923 A210485 A111815 * A210877 A127753 A197736

Adjacent sequences: A281266 A281267 A281268 * A281270 A281271 A281272

Keyword

nonn,tabl

Author

Geoffrey Critzer, Apr 25 2017

Status

approved