A069722 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).

0, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400, 144542561803960320, 1128808577897594880

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Formula

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

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

G.f. for a(n)^2: 1/AGM(1, (1-64*x)^(1/2)). - Benoit Cloitre, Jan 01 2004

a(n) = A059304(n-1), n>1. [R. J. Mathar, Sep 29 2008]

a(n) ~ 2^(3*n-3)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 28 2019

E.g.f.: x * (exp(4*x) * (BesselI(0,4*x) - BesselI(1,4*x)) - 1). - Ilya Gutkovskiy, Nov 03 2021

From Amiram Eldar, Jan 16 2024: (Start)

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

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

Maple

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

Mathematica

Join[{0}, Table[2^(n-1) Binomial[2n-2, n-1], {n, 2, 20}]] (* Harvey P. Dale, Nov 16 2011 *)

Prog

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

Crossrefs

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

Sequence in context: A272865 A084130 A059304 * A343842 A027079 A213441

Adjacent sequences: A069719 A069720 A069721 * A069723 A069724 A069725

Keyword

easy,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved