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 A162551 #58 Oct 23 2023 10:29:48
%S A162551 0,2,8,30,112,420,1584,6006,22880,87516,335920,1293292,4992288,
%T A162551 19315400,74884320,290845350,1131445440,4407922860,17194993200,
%U A162551 67156001220,262564816800,1027583214840,4025232800160,15780742227900,61915399071552
%N A162551 a(n) = 2 * C(2*n,n-1).
%C A162551 Total length of all Dyck paths of length 2n.
%C A162551 a(n) equals the diagonal element A(n,n) of matrix A whose element A(i,j) = A(i-1,j) + A(i,j-1). - _Carmine Suriano_, May 10 2010
%C A162551 a(n) is also the number of solid (3 dimensions) standard Young tableaux of shape [[n,n],[1]]. - _Thotsaporn Thanatipanonda_, Feb 27 2012
%C A162551 With offset = 1, a(n) is the total number of nodes over all binary trees with one child internal and one child external. - _Geoffrey Critzer_, Feb 23 2013
%C A162551 Central terms of the triangle in A051601. - _Reinhard Zumkeller_, Aug 05 2013
%C A162551 a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an odd number of times. Details can be found in Section 4.2 in Pan and Remmel's link. - _Ran Pan_, Feb 01 2016
%C A162551 a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an odd number of times. Details can be found in Section 4.3 in Pan and Remmel's link. - _Ran Pan_, Feb 01 2016
%D A162551 R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley 1996, page 141.
%H A162551 Vincenzo Librandi, <a href="/A162551/b162551.txt">Table of n, a(n) for n = 0..200</a>
%H A162551 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A162551 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A162551 Valentin Ovsienko, <a href="https://arxiv.org/abs/2111.02553">Shadow sequences of integers, from Fibonacci to Markov and back</a>, arXiv:2111.02553 [math.CO], 2021.
%H A162551 Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016
%H A162551 Ping Sun, <a href="http://dx.doi.org/10.1016/j.disc.2012.09.003">Proof of two conjectures of Petkovsek and Wilf on Gessel walks</a>, Discrete Math, 312(24) (2012), 3649-3655. MR2979494. See Th. 1.1, case 2. - _N. J. A. Sloane_, Nov 07 2012
%F A162551 a(n) = 2*A001791(n). - _R. J. Mathar_, Jul 15 2009
%F A162551 E.g.f.: exp(2*x)*2*(BesselI(1,2*x)). - _Peter Luschny_, Aug 26 2012
%F A162551 O.g.f.: ((1 - 2*x)/(1 - 4*x)^(1/2) - 1)/x - _Geoffrey Critzer_, Feb 23 2013
%F A162551 E.g.f.: 2*Q(0) - 2, where Q(k) = 1 - 2*x/(k + 1 - (k + 1)*(2*k + 3)/(2*k + 3 - (k + 2)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 28 2013
%F A162551 a(n) = binomial(2*n+2, n+1) - A028329(n). - _Ran Pan_, Feb 01 2016
%t A162551 nn=25;Drop[CoefficientList[Series[(1-2x)/(1-4x)^(1/2),{x,0,nn}],x],1] (* _Geoffrey Critzer_, Feb 23 2013 *)
%t A162551 Table[2Binomial[2n,n-1],{n,0,30}] (* _Harvey P. Dale_, Oct 26 2016 *)
%o A162551 (Magma) [2*n*Catalan(n): n in [0..30]]; // _Vincenzo Librandi_, Jul 19 2011
%o A162551 (Haskell)
%o A162551 a162551 n = a051601 (2 * n) n -- _Reinhard Zumkeller_, Aug 05 2013
%o A162551 (PARI) a(n) = 2*binomial(2*n,n-1) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A162551 Cf. A001791, A028329, A162549.
%K A162551 nonn
%O A162551 0,2
%A A162551 _Franklin T. Adams-Watters_, Jul 05 2009