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%I #23 Jan 05 2021 21:30:38
%S 1,2,13,150,2541,57330,1623105,55405350,2216439225,101738006370,
%T 5271938032725,304455567165750,19391501988260325,1350480167457671250,
%U 102096314890336391625,8327231070135771543750,728877648485930118800625
%N Total number of Eulerian circuits in labeled multigraphs with n edges.
%H B. Lass, <a href="http://dx.doi.org/10.1016/S0764-4442(01)02049-3">Démonstration combinatoire de la formule de Harer-Zagier</a>, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160.
%H B. Lass, <a href="http://math.univ-lyon1.fr/~lass/articles/pub3zagier.html">Démonstration combinatoire de la formule de Harer-Zagier</a>, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160.
%H Valery Liskovets, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Liskovets/liskovets4.html">A Note on the Total Number of Double Eulerian Circuits in Multigraphs </a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.5
%F a(n) = (2*n)!/(2^n*n!)(3^(n+1)-1)/(2*(n+1)).
%F E.g.f.: (sqrt(1-2*x)-sqrt(1-6*x))/(2*x).
%F From _Sergei N. Gladkovskii_, Jul 25 2012: (Start)
%F G.f.: 1 + 8*x/(G(0)-8*x); where G(k) = x*(k+1)*(2*k+1)*(9*3^k-1) + (k+2)*(3*3^k-1) - x*((k+2)^2)*(3*3^k-1)*(2*k+3)*(27*3^k-1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step).
%F G.f.: (3/2)*G(0) where G(k) = 1 - 1/(3*3^k - 27*x*(k+1)*(2*k+1)*9^k/(9*x*(2*k+1)*(k+1)*3^k - (k+2)/Q2)); (continued fraction, 3rd kind, 3-step).
%F E.g.f.: (sqrt(1-2*x) - sqrt(1-6*x))/(2*x) = G(0)/(2*x); where G(k) = 1 - 3^k/(1 - x*(2*k-1)/(x*(2*k-1) - 3^k*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step).
%F (End)
%t Table[(2n)!/(2^n n!)(3^(n+1)-1)/(2(n+1)),{n,0,20}] (* _Harvey P. Dale_, Aug 24 2019 *)
%o (PARI) x=xx+O(xx^33); Vec(serlaplace((sqrt(1-2*x)-sqrt(1-6*x))/(2*x))) \\ _Michel Marcus_, Dec 11 2014
%Y Cf. A011781.
%K easy,nonn
%O 0,2
%A _Valery A. Liskovets_, Apr 07 2002
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