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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 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 January 28 15:20 EST 2022. Contains 350657 sequences. (Running on oeis4.)