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A273256
Number of simple labeled graphs on n vertices with at most one nontrivial component and all vertex degrees are even.
0
1, 1, 1, 2, 8, 64, 1014, 32593, 2093589, 268333725, 68714765337, 35183979518038, 36028733659454920, 73786955927716463496, 302231441864128208088266, 2475880062024448199702310129, 40564819165779582804001294004849, 1329227995578862816338009185350962977, 87112285929737129482236375622145146977689
OFFSET
0,4
COMMENTS
Some graph theory texts call these graphs Eulerian. Cf. A033678.
REFERENCES
D. B. West, Introduction to Graph Theory, 2nd edition, Pearson Education, 2001, page 27.
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!.
EXAMPLE
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
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]
CROSSREFS
Sequence in context: A153570 A153533 A153562 * A192414 A153543 A153571
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Aug 28 2016
STATUS
approved