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A005193 a(n) is the number of alpha-labelings of graphs with n edges.

%I A005193 M1231 #57 Oct 28 2023 15:28:42
%S A005193 1,2,4,10,30,106,426,1930,9690,53578,322650,2106250,14790810,
%T A005193 111327178,893091930,7614236170,68695024410,654301474378,
%U A005193 6557096219610,69005893630090,760519875693210,8763511069234378,105343011537811290,1319139904954848010
%N A005193 a(n) is the number of alpha-labelings of graphs with n edges.
%C A005193 Old name was: Balanced labeled graphs. New name taken from Mar 06 2021 comment from _Don Knuth_.
%D A005193 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005193 C. Barrientos and S. M. Minion, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Minion/minion3.pdf">Enumerating families of labeled graphs</a>, J. Integer Seq., 18(2015), #15.1.7.
%H A005193 Henryk Fuks and Kate Sullivan, <a href="https://arxiv.org/abs/0711.1349">Enumeration of number-conserving cellular automata rules with two inputs</a>, arXiv:0711.1349 [nlin.CG], 2007; Journal of Cellular Automata 2 vol. 2 pp. 141-148 (2007).
%H A005193 David A. Sheppard, <a href="http://dx.doi.org/10.1016/0012-365X(76)90051-0">The factorial representation of major balanced labelled graphs</a>, Discrete Math., 15(1976), no. 4, 379-388.
%F A005193 If n is even then a(n) = 2*Sum_{j=1..floor(n/2)} j!^2*j^(n-2*j), otherwise a(n) = 2*Sum_{j=1..floor(n/2)} j!^2*j^(n-2*j) + ((n+1)/2)!*((n-1)/2)!. - _Jonathan Vos Post_, Nov 13 2007
%p A005193 A005193 := proc(q)
%p A005193     2*add((j!)^2*j^(q-2*j),j=1..q/2) ;
%p A005193     if type(q,'odd') then
%p A005193         %+((q+1)/2)!*((q-1)/2)! ;
%p A005193     else
%p A005193         % ;
%p A005193     end if;
%p A005193 end proc:
%p A005193 seq(A005193(n),n=1..40) ; # _R. J. Mathar_, Jul 13 2014
%t A005193 a[n_] := 2 Sum[(j!)^2*j^(n-2j), {j, 1, n/2}] + Boole[OddQ[n]]*((n+1)/2)! * ((n-1)/2)!;
%t A005193 Array[a, 24] (* _Jean-François Alcover_, Nov 20 2017 *)
%Y A005193 Cf. A034384, A342225.
%K A005193 nonn
%O A005193 1,2
%A A005193 _N. J. A. Sloane_
%E A005193 Renamed (using Comments entry from _Don Knuth_) by _Jon E. Schoenfield_, Oct 28 2023