A069720 a(n) = 2^(n-1)*binomial(2n-1, n).

1, 6, 40, 280, 2016, 14784, 109824, 823680, 6223360, 47297536, 361181184, 2769055744, 21300428800, 164317593600, 1270722723840, 9848101109760, 76467608616960, 594748067020800, 4632774416793600, 36135640450990080, 282202144474398720, 2206307674981662720, 17266755717247795200

Offset

1,2

Comments

Number of rooted unicursal planar maps with n edges (unicursal means that exactly two nodes are of odd valency; there is an Eulerian path).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Harlan J. Brothers, Pascal's Prism: Supplementary Material.

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

Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Formula

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

G.f.: (1-sqrt(1-8x))/(4x*sqrt(1-8x)) = 2/(sqrt(1-8x)(1-sqrt(1-8x)))-1/(2x). - Paul Barry, Sep 06 2004

D-finite with recurrence n*a(n) + 4*(1-2n)*a(n-1) = 0. - R. J. Mathar, Apr 01 2012

E.g.f.: a(n) = n! * [x^n] (exp(4*x)*BesselI(0, 4*x) - 1)/4. - Peter Luschny, Aug 25 2012

a(n) = A000079(n-1) * A001700(n-1); for n > 1: a(n) = 2*A082143(n-1). - Reinhard Zumkeller, Jan 15 2015

From Amiram Eldar, Jan 16 2024: (Start)

Sum_{n>=1} 1/a(n) = 4/7 + 32*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).

Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + 16*log(2)/27. (End)

Maple

Z:=(1-sqrt(1-2*z))*4^(n-1)/sqrt(1-2*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..20); # Zerinvary Lajos, Jan 01 2007

Mathematica

Table[2^(n-1) Binomial[2n-1, n], {n, 20}] (* Harvey P. Dale, Jan 20 2013 *)

Prog

(Haskell)
a069720 n = (a000079 $ n - 1) * (a001700 $ n - 1)
-- Reinhard Zumkeller, Jan 15 2015
(PARI) a(n) = binomial(2*n-1, n)<<(n-1) \\ Charles R Greathouse IV, Feb 06 2017
(Magma) [2^(n-2)*Binomial(2*n, n): n in [1..25]]; // Vincenzo Librandi, Apr 14 2018

Crossrefs

First superdiagonal of number array A082137.

Cf. A069724, A003584, A069723, A082143, A000079, A001700.

Sequence in context: A122471 A178397 A090041 * A005037 A081337 A316912

Adjacent sequences: A069717 A069718 A069719 * A069721 A069722 A069723

Keyword

easy,nice,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved