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 A069736 Total number of Eulerian circuits in labeled multigraphs with n edges. 1
 1, 2, 13, 150, 2541, 57330, 1623105, 55405350, 2216439225, 101738006370, 5271938032725, 304455567165750, 19391501988260325, 1350480167457671250, 102096314890336391625, 8327231070135771543750, 728877648485930118800625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160. B. Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333 (2001) No 3, 155-160. Valery Liskovets, A Note on the Total Number of Double Eulerian Circuits in Multigraphs , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.5 FORMULA a(n) = (2*n)!/(2^n*n!)(3^(n+1)-1)/(2*(n+1)). E.g.f.: (sqrt(1-2*x)-sqrt(1-6*x))/(2*x). From Sergei N. Gladkovskii, Jul 25 2012 (Start) 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). 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). 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). (End) PROG (PARI) x=xx+O(xx^33); Vec(serlaplace((sqrt(1-2*x)-sqrt(1-6*x))/(2*x))) \\ Michel Marcus, Dec 11 2014 CROSSREFS Cf. A011781. Sequence in context: A079330 A059367 A204554 * A058192 A054382 A062593 Adjacent sequences:  A069733 A069734 A069735 * A069737 A069738 A069739 KEYWORD easy,nonn AUTHOR Valery A. Liskovets, Apr 07 2002 STATUS approved

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