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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..18.

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

Adjacent sequences:  A273253 A273254 A273255 * A273257 A273258 A273259

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Aug 28 2016

STATUS

approved

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Last modified June 19 03:34 EDT 2018. Contains 305572 sequences. (Running on oeis4.)