A069722 - OEIS

%I #35 Jan 16 2024 01:39:44

%S 0,4,24,160,1120,8064,59136,439296,3294720,24893440,189190144,

%T 1444724736,11076222976,85201715200,657270374400,5082890895360,

%U 39392404439040,305870434467840,2378992268083200,18531097667174400,144542561803960320,1128808577897594880

%N Number of rooted unicursal planar maps with n edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

%H Vincenzo Librandi, <a href="/A069722/b069722.txt">Table of n, a(n) for n = 1..200</a>

%H 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.

%F a(n) = 2^(n-1)*binomial(2n-2, n-1), n>1.

%F a(n) = 2*A069723(n), n>1.

%F G.f. for a(n)^2: 1/AGM(1, (1-64*x)^(1/2)). - _Benoit Cloitre_, Jan 01 2004

%F a(n) = A059304(n-1), n>1. [_R. J. Mathar_, Sep 29 2008]

%F a(n) ~ 2^(3*n-3)/sqrt(Pi*n). - _Vaclav Kotesovec_, Sep 28 2019

%F E.g.f.: x * (exp(4*x) * (BesselI(0,4*x) - BesselI(1,4*x)) - 1). - _Ilya Gutkovskiy_, Nov 03 2021

%F From _Amiram Eldar_, Jan 16 2024: (Start)

%F Sum_{n>=2} 1/a(n) = 1/7 + 8*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).

%F Sum_{n>=2} (-1)^n/a(n) = 1/9 + 4*log(2)/27. (End)

%p Z:=(1-sqrt(1-z))*8^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=0..19); # _Zerinvary Lajos_, Jan 01 2007

%t Join[{0},Table[2^(n-1) Binomial[2n-2,n-1],{n,2,20}]] (* _Harvey P. Dale_, Nov 16 2011 *)

%o (Magma) [0] cat[2^(n-1)*Binomial(2*n-2, n-1): n in [2..20]]; // _Vincenzo Librandi_, Nov 17 2011

%Y Cf. A059304, A069720, A069721, A069723, A089156.

%K easy,nonn

%O 1,2

%A _Valery A. Liskovets_, Apr 07 2002