A069723 a(n) = 2^(n-1)*binomial(2*n-3, n-1).

1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360, 12446720, 94595072, 722362368, 5538111488, 42600857600, 328635187200, 2541445447680, 19696202219520, 152935217233920, 1189496134041600, 9265548833587200, 72271280901980160, 564404288948797440, 4412615349963325440

Offset

1,2

Comments

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

Links

G. C. Greubel, 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.

Formula

a(n) = A069722(n)/2, n>1.

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

With offset 0: a(n) = (0^n + 2^n*binomial(2n, n))/2. - Paul Barry, Sep 24 2004

D-finite with recurrence (-n+1)*a(n) + 4*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012

With offset 0: a(n) = 2^n*rf(n,n)/n! = 2^n*A088218(n), where rf denotes the rising factorial. - Peter Luschny, Nov 30 2014

a(n) = Sum_{k=0..n} binomial(n+k-1,k)*binomial(2*n-1, n-k). - Vladimir Kruchinin, Nov 11 2016

a(n) ~ 2^(3*n-4)/sqrt(Pi*n). - Ilya Gutkovskiy, Nov 11 2016

From Amiram Eldar, Jan 16 2024: (Start)

Sum_{n>=1} 1/a(n) = 9/7 + 16*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).

Sum_{n>=1} (-1)^(n+1)/a(n) = 7/9 - 8*log(2)/27. (End)

Maple

Z:=(1-sqrt(1-z))*8^n/sqrt(1-z)/2: Zser:=series(Z, z=0, 33): seq(coeff(Zser, z, n), n=0..20); # Zerinvary Lajos, Jan 16 2007

Mathematica

Table[2^(n - 1) * Binomial[2*n - 3, n - 1], {n, 1, 50}] (* G. C. Greubel, Jan 15 2017 *)

Prog

(Sage)
# Assuming offset 0:
A069723  = lambda n: (rising_factorial(n, n)/factorial(n)) << n
[A069723(n) for n in (0..20)] # Peter Luschny, Nov 30 2014

Crossrefs

Main diagonal of array A082137.

Cf. A069720, A069721, A069722, A082143, A082144, A082145, A088218.

Sequence in context: A240836 A270919 A082142 * A063481 A274782 A185020

Adjacent sequences: A069720 A069721 A069722 * A069724 A069725 A069726

Keyword

easy,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved