This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069720 #57 Jan 23 2025 00:16:12
%S A069720 1,6,40,280,2016,14784,109824,823680,6223360,47297536,361181184,
%T A069720 2769055744,21300428800,164317593600,1270722723840,9848101109760,
%U A069720 76467608616960,594748067020800,4632774416793600,36135640450990080,282202144474398720,2206307674981662720,17266755717247795200
%N A069720 a(n) = 2^(n-1)*binomial(2*n-1, n).
%C A069720 Number of rooted unicursal planar maps with n edges (unicursal means that exactly two nodes are of odd valency; there is an Eulerian path).
%H A069720 Reinhard Zumkeller, <a href="/A069720/b069720.txt">Table of n, a(n) for n = 1..1000</a>
%H A069720 Harlan J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.
%H A069720 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.
%H A069720 Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="http://dx.doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091.
%F A069720 a(n) = 2^(n-2)*binomial(2*n, n).
%F A069720 G.f.: (1-sqrt(1-8*x))/(4*x*sqrt(1-8*x)) = 2/(sqrt(1-8*x)*(1-sqrt(1-8*x))) - 1/(2*x). - _Paul Barry_, Sep 06 2004
%F A069720 D-finite with recurrence: n*a(n) - 4*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Apr 01 2012
%F A069720 E.g.f.: a(n) = n! * [x^n] (exp(4*x)*BesselI(0, 4*x) - 1)/4. - _Peter Luschny_, Aug 25 2012
%F A069720 From _Reinhard Zumkeller_, Jan 15 2015: (Start)
%F A069720 a(n) = A000079(n-1) * A001700(n-1); for n > 1:
%F A069720 a(n) = 2*A082143(n-1). (End)
%F A069720 From _Amiram Eldar_, Jan 16 2024: (Start)
%F A069720 Sum_{n>=1} 1/a(n) = 4/7 + 32*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
%F A069720 Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + 16*log(2)/27. (End)
%F A069720 a(n) = ((2*n)!/4) * [x^n] (BesselI(0, 2*sqrt(2*x)) - 1). - _G. C. Greubel_, Jan 18 2025
%p A069720 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
%t A069720 Table[2^(n-1) Binomial[2n-1,n],{n,20}] (* _Harvey P. Dale_, Jan 20 2013 *)
%o A069720 (Haskell)
%o A069720 a069720 n = (a000079 $ n - 1) * (a001700 $ n - 1)
%o A069720 -- _Reinhard Zumkeller_, Jan 15 2015
%o A069720 (PARI) a(n) = binomial(2*n-1,n)<<(n-1) \\ _Charles R Greathouse IV_, Feb 06 2017
%o A069720 (Magma) [2^(n-2)*Binomial(2*n, n): n in [1..25]]; // _Vincenzo Librandi_, Apr 14 2018
%o A069720 (SageMath)
%o A069720 def A069720(n): return 2^(n-2)*binomial(2*n, n)
%o A069720 print([A069720(n) for n in range(1,31)]) # _G. C. Greubel_, Jan 18 2025
%Y A069720 First superdiagonal of number array A082137.
%Y A069720 Cf. A000079, A001700, A003584, A069723, A069724, A082143.
%K A069720 easy,nice,nonn
%O A069720 1,2
%A A069720 _Valery A. Liskovets_, Apr 07 2002