A162549 -id:A162549 - cp's OEIS frontend

A162551 a(n) = 2 * C(2*n,n-1).

0, 2, 8, 30, 112, 420, 1584, 6006, 22880, 87516, 335920, 1293292, 4992288, 19315400, 74884320, 290845350, 1131445440, 4407922860, 17194993200, 67156001220, 262564816800, 1027583214840, 4025232800160, 15780742227900, 61915399071552

Offset: 0

Franklin T. Adams-Watters, Jul 05 2009

nonn

Total length of all Dyck paths of length 2n.
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
a(n) is also the number of solid (3 dimensions) standard Young tableaux of shape [[n,n],[1]]. - Thotsaporn Thanatipanonda, Feb 27 2012
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
Central terms of the triangle in A051601. - Reinhard Zumkeller, Aug 05 2013
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
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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Valentin Ovsienko, Shadow sequences of integers, from Fibonacci to Markov and back, arXiv:2111.02553 [math.CO], 2021.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016
Ping Sun, Proof of two conjectures of Petkovsek and Wilf on Gessel walks, Discrete Math, 312(24) (2012), 3649-3655. MR2979494. See Th. 1.1, case 2. - _N. J. A. Sloane_, Nov 07 2012

Cf. A001791, A028329, A162549.

a162551 n = a051601 (2 * n) n  -- Reinhard Zumkeller, Aug 05 2013

[2*n*Catalan(n): n in [0..30]]; // Vincenzo Librandi, Jul 19 2011

nn=25;Drop[CoefficientList[Series[(1-2x)/(1-4x)^(1/2),{x,0,nn}],x],1]  (* Geoffrey Critzer, Feb 23 2013 *)
Table[2Binomial[2n,n-1],{n,0,30}] (* Harvey P. Dale, Oct 26 2016 *)

a(n) = 2*binomial(2*n,n-1) \\ Charles R Greathouse IV, Oct 23 2023

a(n) = 2*A001791(n). - R. J. Mathar, Jul 15 2009
E.g.f.: exp(2*x)*2*(BesselI(1,2*x)). - Peter Luschny, Aug 26 2012
O.g.f.: ((1 - 2*x)/(1 - 4*x)^(1/2) - 1)/x - Geoffrey Critzer, Feb 23 2013
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
a(n) = binomial(2*n+2, n+1) - A028329(n). - Ran Pan, Feb 01 2016

A162550 2n repeated C_n times, where C_n = A000108(n) is a Catalan number.

Original entry on oeis.org

0, 2, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

Franklin T. Adams-Watters, Jul 05 2009

nonn

Lengths of Dyck paths (A162549).

Cf. A162549, A072643, A000108.

a(n) = 2 * A072643(n).

cp's OEIS Frontend

A162551 a(n) = 2 * C(2*n,n-1).

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