A162551 - OEIS

%I #58 Oct 23 2023 10:29:48

%S 0,2,8,30,112,420,1584,6006,22880,87516,335920,1293292,4992288,

%T 19315400,74884320,290845350,1131445440,4407922860,17194993200,

%U 67156001220,262564816800,1027583214840,4025232800160,15780742227900,61915399071552

%N a(n) = 2 * C(2*n,n-1).

%C Total length of all Dyck paths of length 2n.

%C 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 a(n) is also the number of solid (3 dimensions) standard Young tableaux of shape [[n,n],[1]]. - _Thotsaporn Thanatipanonda_, Feb 27 2012

%C 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 Central terms of the triangle in A051601. - _Reinhard Zumkeller_, Aug 05 2013

%C 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 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 R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley 1996, page 141.

%H Vincenzo Librandi, <a href="/A162551/b162551.txt">Table of n, a(n) for n = 0..200</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H 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 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 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 a(n) = 2*A001791(n). - _R. J. Mathar_, Jul 15 2009

%F E.g.f.: exp(2*x)*2*(BesselI(1,2*x)). - _Peter Luschny_, Aug 26 2012

%F O.g.f.: ((1 - 2*x)/(1 - 4*x)^(1/2) - 1)/x - _Geoffrey Critzer_, Feb 23 2013

%F 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 a(n) = binomial(2*n+2, n+1) - A028329(n). - _Ran Pan_, Feb 01 2016

%t nn=25;Drop[CoefficientList[Series[(1-2x)/(1-4x)^(1/2),{x,0,nn}],x],1]  (* _Geoffrey Critzer_, Feb 23 2013 *)

%t Table[2Binomial[2n,n-1],{n,0,30}] (* _Harvey P. Dale_, Oct 26 2016 *)

%o (Magma) [2*n*Catalan(n): n in [0..30]]; // _Vincenzo Librandi_, Jul 19 2011

%o (Haskell)

%o a162551 n = a051601 (2 * n) n  -- _Reinhard Zumkeller_, Aug 05 2013

%o (PARI) a(n) = 2*binomial(2*n,n-1) \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Cf. A001791, A028329, A162549.

%K nonn

%O 0,2

%A _Franklin T. Adams-Watters_, Jul 05 2009