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 A063016 #50 Nov 01 2022 13:48:12
%S A063016 1,3,14,75,434,2646,16764,109395,730730,4973826,34381412,240728670,
%T A063016 1703826292,12170930700,87633375480,635351667075,4634365164570,
%U A063016 33985474184970,250419761106900,1853107999454250,13765951702923420,102618937160787060,767411273728449480
%N A063016 a(n) is the product of Catalan(n) and (2^(n+1) - 1).
%C A063016 From _Michael Wallner_, Nov 03 2021: (Start)
%C A063016 a(n) is also the number of n X 2 Young tableaux with (possibly) vertical walls. The entries in cells that are separated by such a wall do not have to obey any order constraints. See Banderier, Wallner 2021 and Banderier et al. 2018.
%C A063016 a(n) is also the number of binary trees with n vertices and marked leaves, where at least 1 leaf has to be marked. Banderier, Wallner 2021 give a bijection to n X 2 Young tableaux with vertical walls. (End)
%H A063016 Harry J. Smith, <a href="/A063016/b063016.txt">Table of n, a(n) for n = 0..200</a>
%H A063016 C. Banderier, P. Marchal, and M. Wallner, <a href="http://ceur-ws.org/Vol-2113/paper5.pdf">Rectangular Young tableaux with local decreases and the density method for uniform random generation</a>, GASCOM 2018. Vol. 2113. CEUR Workshop Proceedings, 2018.
%H A063016 C. Banderier and M. Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47Banderier.pdf">Young tableaux with periodic walls: counting with the density method</a> Sém. Lothar. Combin. 85B (2021), Art. 47, 12 pp.
%F A063016 a(n) = Catalan(n)*(2^(n+1) - 1) = A000108(n) * A126646(n).
%F A063016 D-finite with recurrence: a(n) = 2*(2*n-1)*(3*n*a(n-1)-4*(2*n-3)*a(n-2))/((n+1)*n). - _Georg Fischer_, Jun 06 2021
%F A063016 G.f.: A(x) = (sqrt(1-4*x) - sqrt(1-8*x))/(2*x).
%F A063016 G.f.: G(0)/(2*x) where G(k) = 1 - 2^k/(1 - 2*x*(2*k-1)/(2*x*(2*k-1) - 2^k*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Jul 24 2012
%F A063016 From _Peter Bala_, Aug 17 2021: (Start)
%F A063016 a(n) = Sum_{k = 0..n} A046521(n,k)*Catalan(k).
%F A063016 G.f.: A(x) = 1/sqrt(1 - 4*x)*c(x/(1 - 4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Inversely, c(x) = 1/sqrt(1 + 4*x)*A(x/(1 + 4*x)).
%F A063016 Series reversion of x*A(x) = x*(1 - 3*x + 4*x^2*c(-2*x^2)). (End)
%t A063016 Table[CatalanNumber[n]*(2^(n+1)-1),{n,0,20}] (* _Harvey P. Dale_, Oct 20 2014 *)
%o A063016 (PARI) a(n) = (2^(n + 1) - 1)*binomial(2*n, n)/(n + 1); \\ _Harry J. Smith_, Aug 16 2009
%o A063016 (Sage)
%o A063016 def A063016(n) :
%o A063016 return (8^(n+1)-4^(n+1))*factorial(n-1/2)/(4*sqrt(pi)*factorial(n+1))
%o A063016 [A063016(i) for i in (0..20)] # _Peter Luschny_, Jul 24 2012
%Y A063016 Cf. A000108, A063017, A126646, A007317, A046521.
%K A063016 nonn,easy
%O A063016 0,2
%A A063016 _Olivier Gérard_, Jul 04 2001
%E A063016 Initial term 0 removed by _Harry J. Smith_, Aug 16 2009