A212553
Irreducible near matchings over 2n+1 vertices.
Original entry on oeis.org
3, 15, 126, 1395, 18678, 289926, 5090220, 99466371, 2138575590, 50153137650, 1274013771300, 34853264146350, 1021795130252988, 31964591332833900, 1062945070050997656, 37447053359058465507, 1393372283171378001030
Offset: 1
a(1) corresponds to the 3-vertex graphs with edge sets {{1,2},{2,3}}, {{1,2},{1,3}} and {{1,3},{2,3}} respectively.
a(2) corresponds to the 5-vertex graphs with edge sets {{1,2},{2,4},{3,5}}, {{1,4},{2,3},{2,5}} ...
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A000699(n)={my(A); A=O(x); for(i=1, n, A=x+A*(2*x*A'-A)); polcoeff(A, n)}
a(n)=A000699(n+1)*(2*n+1)*(n+1)/(2*n);
vector(66,n,a(n))
/* Joerg Arndt, May 27 2012 */
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.
Original entry on oeis.org
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
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;
First positive term in each even row is
A001147.
First positive term in each odd row is
A200142.
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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
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