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%I #74 Feb 11 2022 04:48:35
%S 1,3,27,405,8505,229635,7577955,295540245,13299311025,678264862275,
%T 38661097149675,2435649120429525,168059789309637225,
%U 12604484198222791875,1020963220056046141875,88823800144876014343125,8260613413473469333910625,817800727933873464057151875
%N Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+3).
%C Total number of Eulerian circuits in rooted labeled multigraphs with n edges. - _Valery A. Liskovets_, Apr 07 2002
%C Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the east quadrant {(x,y): x >= |y|} and using steps (0,1), (0,-1), (1,1), (-1,-1), and (1,0). - _Alois P. Heinz_, Oct 13 2016
%H Vincenzo Librandi, <a href="/A011781/b011781.txt">Table of n, a(n) for n = 0..200</a>
%H Fatemeh Bagherzadeh, M. Bremner, and S. Madariaga, <a href="http://arxiv.org/abs/1611.01214">Jordan Trialgebras and Post-Jordan Algebras</a>, arXiv:1611.01214 [math.RA], 2016.
%H Murray Bremner and Martin Markl, <a href="https://arxiv.org/abs/1809.08191">Distributive laws between the Three Graces</a>, arXiv:1809.08191 [math.AT], 2018.
%H Bodo 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(3) (2001), 155-160.
%H Bodo Lass, <a href="https://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, Vol. 333, No. 3 (2001), pp. 155-160; <a href="http://math.univ-lyon1.fr/~lass/articles/pub3zagier.html">alternative link</a>.
%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 E.g.f.: (1-6*x)^(-1/2).
%F a(n) = 3^n*(2*n-1)!!.
%F G.f.: 1/(1-3*x/(1-6*x/(1-9*x/(1-12*x/(1-15*x/(1-18*x/(1-21*x/(1-24*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F a(n) = (-3)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. [_Mircea Merca_, May 03 2012]
%F G.f.: T(0), where T(k) = 1 - 3*x*(k+1)/( 3*x*(k+1) - 1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 24 2013
%F a(n) = 6^n * gamma(n + 1/2) / sqrt(Pi). - _Daniel Suteu_, Jan 06 2017
%F 0 = a(n)*(+6*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) and a(n) = (-1)^n / a(-n) for all n in Z. - _Michael Somos_, Jan 06 2017
%F D-finite with recurrence: a(n) +3*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 20 2018
%F From _Amiram Eldar_, Feb 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*sqrt(Pi/6)*erf(1/sqrt(6)), where erf is the error function.
%F Sum_{n>=0} (-1)^n/a(n) = 1 - exp(-1/6)*sqrt(Pi/6)*erfi(1/sqrt(6)), where erfi is the imaginary error function. (End)
%e G.f. = 1 + 3*x + 27*x^2 + 405*x^3 + 8505*x^4 + 229635*x^5 + 7577955*x^6 + ...
%t Table[Product[6k+3,{k,0,n-1}],{n,0,20}] (* or *) Table[6^(n-1) Pochhammer[ 1/2,n-1],{n,21}] (* _Harvey P. Dale_, May 09 2012 *)
%t Table[6^n*Pochhammer[1/2, n], {n,0,20}] (* _G. C. Greubel_, Aug 20 2019 *)
%o (PARI) {a(n) = if( n<0, (-1)^n / a(-n), (3/2)^n * (2*n)! / n!)}; /* _Michael Somos_, Feb 10 2002, revised and extended _Michael Somos_, Jan 06 2017 */
%o (Magma) [(3/2)^n*Factorial(2*n)/Factorial(n):n in [0..20]]; // _Vincenzo Librandi_, May 09 2012
%o (Sage) [6^n*rising_factorial(1/2, n) for n in (0..20)] # _G. C. Greubel_, Aug 20 2019
%o (GAP) F:=Factorial;; List([0..20], n-> (3/2)^n*(F(2*n)/F(n)) ); # _G. C. Greubel_, Aug 20 2019
%Y Cf. A001147, A035309, A047657, A048994, A049308, A069736, A273464, A277358.
%K nonn,easy
%O 0,2
%A Lee D. Killough (killough(AT)wagner.convex.com)
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