A273256 Number of simple labeled graphs on n vertices with at most one nontrivial component and all vertex degrees are even.

1, 1, 1, 2, 8, 64, 1014, 32593, 2093589, 268333725, 68714765337, 35183979518038, 36028733659454920, 73786955927716463496, 302231441864128208088266, 2475880062024448199702310129, 40564819165779582804001294004849, 1329227995578862816338009185350962977, 87112285929737129482236375622145146977689

Offset: 0

Author

Geoffrey Critzer, Aug 28 2016

Keywords

• nonn

Comments

Some graph theory texts call these graphs Eulerian. Cf. A033678.

Examples

			a(4) = 8 because there are 1+4+3=8 labelings on these three graphs
1)
o o
o o
2)
o-o
|/
o o
3)
o-o
| |
o-o
		

References

• D. B. West, Introduction to Graph Theory, 2nd edition, Pearson Education, 2001, page 27.

Programs

Mathematica

nn = 18; Clear[g];g[z_] := Sum[2^Binomial[n - 1, 2] z^n/n!, {n, 1, nn}];Range[0, nn]! CoefficientList[Series[Exp[z] (Log[g[z] + 1] - z + 1), {z, 0, nn}], z]

Formula

E.g.f.: exp(x)*(log(A(x) + 1) - x + 1) where A(x) = Sum_{n>=1} 2^binomial(n-1,2)x^n/n!.