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%I #52 Jan 16 2024 08:33:37
%S 1,2,12,80,560,4032,29568,219648,1647360,12446720,94595072,722362368,
%T 5538111488,42600857600,328635187200,2541445447680,19696202219520,
%U 152935217233920,1189496134041600,9265548833587200,72271280901980160,564404288948797440,4412615349963325440
%N a(n) = 2^(n-1)*binomial(2*n-3, n-1).
%C 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).
%H G. C. Greubel, <a href="/A069723/b069723.txt">Table of n, a(n) for n = 1..1000</a>
%H Harlan J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal%27s_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</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) = A069722(n)/2, n>1.
%F G.f.: 4*x/(sqrt(1-8*x) * (1-sqrt(1-8*x))). - _Paul Barry_, Sep 06 2004
%F With offset 0: a(n) = (0^n + 2^n*binomial(2n, n))/2. - _Paul Barry_, Sep 24 2004
%F D-finite with recurrence (-n+1)*a(n) + 4*(2*n-3)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012
%F 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
%F a(n) = Sum_{k=0..n} binomial(n+k-1,k)*binomial(2*n-1, n-k). - _Vladimir Kruchinin_, Nov 11 2016
%F a(n) ~ 2^(3*n-4)/sqrt(Pi*n). - _Ilya Gutkovskiy_, Nov 11 2016
%F From _Amiram Eldar_, Jan 16 2024: (Start)
%F Sum_{n>=1} 1/a(n) = 9/7 + 16*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 7/9 - 8*log(2)/27. (End)
%p 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
%t Table[2^(n - 1) * Binomial[2*n - 3, n - 1], {n, 1,50}] (* _G. C. Greubel_, Jan 15 2017 *)
%o (Sage)
%o # Assuming offset 0:
%o A069723 = lambda n: (rising_factorial(n, n)/factorial(n)) << n
%o [A069723(n) for n in (0..20)] # _Peter Luschny_, Nov 30 2014
%Y Main diagonal of array A082137.
%Y Cf. A069720, A069721, A069722, A082143, A082144, A082145, A088218.
%K easy,nonn
%O 1,2
%A _Valery A. Liskovets_, Apr 07 2002
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