%I M3146 #35 Feb 20 2021 16:10:47
%S 1,0,1,3,38,720,26614,1858122,250586792,66121926720,34442540326456,
%T 35611003057733928,73321307277341501168,301201690357187097528960,
%U 2471354321681605983102370864,40525241311304939167532163726672
%N Number of labeled Eulerian graphs with n nodes.
%C From _Petros Hadjicostas_, Feb 20 2021: (Start)
%C See the comments for A058878 about the different (and sometimes confusing) terminology regarding even and (connected or not) Euler graphs.
%C Cao (2002) uses the term "connected labeled Eulerian graphs" in the title of his Section 4.3, where this sequence appears, and the term "labeled Eulerian graph" in some of the discussion of that section. The author does cite the definition of Harary and Palmer (1973) for an Euler or Eulerian graph (as a connected even graph).
%C Note that all graphs counted by this sequence, by A058878, and by the triangular arrays A228550 and A341743 are assumed to be simple (with no loops and no multiple edges). Read (1962), however, indicates how to solve similar counting problems in the case of graphs with loops and/or multiple edges. (End)
%D F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 12, Eq. (1.4.6).
%D E. M. Palmer in L. W. Beineke and R. J. Wilson, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 385ff.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A033678/b033678.txt">Table of n, a(n) for n=1..50</a>
%H Huantian Cao, <a href="http://cobweb.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002.
%H Huantian Cao, <a href="/A000009/a000009.pdf">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002 [Local copy, with permission]
%H math.stackexchange.com, <a href="https://math.stackexchange.com/questions/1414667/is-it-possible-disconnected-graph-has-euler-circuit">Is it possible disconnected graph has euler circuit? [sic]</a>, August 30, 2015.
%H Ronald C. Read, <a href="https://doi.org/10.4153/CJM-1962-039-0">Euler graphs on labelled nodes</a>, Canadian Journal of Mathematics, 14 (1962), 482-486; see the discussion in Section 4, following Eq. (8) on p. 486.
%p A033678 := proc(n) option remember; local k; if n=1 then 1 else 2^binomial(n-1,2)-(1/n)*add(k*binomial(n,k)*2^binomial(n-k-1,2)*A033678(k),k=1..n-1); fi; end;
%t n = 16; (Series[ Log[ 1 + Sum[ 2^( (p-1)(p-2)/2 )x^p/(p!), {p, 1, n} ] ], {x, 0, n} ] // CoefficientList[#, x]& // Rest) * Range[n]! (* truncated exponential generating function *)
%t (* Second program: *)
%t a[n_] := a[n] = If[n == 1, 1, 2^Binomial[n-1, 2]-(1/n)*Sum[k*Binomial[n, k]*2^Binomial[n-k-1, 2]*a[k], {k, 1, n-1}]]; Table[a[n], {n, 1, 16}] (* _Jean-François Alcover_, Feb 11 2014, after Maple *)
%o (Sage)
%o @cached_function
%o def A033678(n):
%o if n == 1: return 1
%o return 2^binomial(n-1, 2)-sum(k*2^((k-n+1)*(k-n+2)/2)*binomial(n,k)*A033678(k) for k in (1..n-1))/n
%o [A033678(n) for n in (1..16)] # _Peter Luschny_, Jan 17 2016
%Y Cf. A228550 (with multiple components)
%Y Row sums of A341743.
%K easy,nonn,nice
%O 1,4
%A _N. J. A. Sloane_ and Geoffrey Mess (mess(AT)math.ucla.edu)