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A069721 Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

%I A069721 #29 Jan 16 2024 01:39:40
%S A069721 0,0,4,40,336,2688,21120,164736,1281280,9957376,77395968,601968640,
%T A069721 4686094336,36515020800,284817162240,2223764766720,17379001958400,
%U A069721 135942415319040,1064286014668800,8338993950228480,65388301768458240,513094808135270400,4028909667357818880
%N A069721 Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).
%H A069721 Robert Israel, <a href="/A069721/b069721.txt">Table of n, a(n) for n = 1..1109</a>
%H A069721 Valery A. Liskovets and Timothy R. S. Walsh, <a href="http://dx.doi.org/10.1016/j.disc.2003.09.015">Enumeration of Eulerian and unicursal planar maps</a>, Discr. Math., 282 (2004), 209-221.
%H A069721 Youngja Park and SeungKyung Park, <a href="http://dx.doi.org/10.1016/j.disc.2016.04.024">Enumeration of generalized lattice paths by string types, peaks, and ascents</a>, Discrete Mathematics 339.11 (2016): 2652-2659.
%F A069721 a(n) = 2^(n-2)*(n-2)*binomial(2n-2, n-1)/n, n>1.
%F A069721 From _Robert Israel_, Nov 12 2016: (Start)
%F A069721 G.f.: 32*x^3/(sqrt(1-8*x)*(1+sqrt(1-8*x))^3).
%F A069721 E.g.f.: ((1-6*x)/4)*exp(4*x)*I_0(4*x)+(3/2)*exp(4*x)*I_1(4*x)+x/2-1/4, where I_0 and I_1 are modified Bessel functions of the first kind.
%F A069721 a(n+1) = (4*(n-1)*(2*n-1)/((n+1)*(n-2)))*a(n).
%F A069721 a(n) ~ 8^n/(16*sqrt(Pi*n)). (End)
%F A069721 From _Amiram Eldar_, Jan 16 2024: (Start)
%F A069721 Sum_{n>=3} 1/a(n) = 11/14 - 26*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
%F A069721 Sum_{n>=3} (-1)^(n+1)/a(n) = 37*log(2)/27 - 13/18. (End)
%e A069721 G.f. = 4*x^3 + 40*x^4 + 336*x^5 + 2688*x^6 + 21120*x^7 + 164736*x^8 + ...
%p A069721 0, seq(2^(n-2)*(n-2)*binomial(2*n-2, n-1)/n, n=2..30); # _Robert Israel_, Nov 12 2016
%t A069721 a[ n_] := SeriesCoefficient[ ((1 - Sqrt[1 - 8 x])/2)^3 / (2 Sqrt[1 - 8 x] ), {x, 0, n}]; (* _Michael Somos_, Nov 13 2016 *)
%o A069721 (Magma) [0] cat [2^(n-2)*(n-2)*Binomial(2*n-2, n-1)/n: n in [2..25]]; // _Vincenzo Librandi_, Nov 13 2016
%Y A069721 Cf. A069720, A069722, A069723.
%K A069721 easy,nonn
%O A069721 1,3
%A A069721 _Valery A. Liskovets_, Apr 07 2002