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

Author

Franklin T. Adams-Watters, Jul 05 2009

Keywords

• nonn

Comments

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

References

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

Links

• 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

Crossrefs

Cf. A001791, A028329, A162549.

Programs

Haskell

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

Magma

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

Mathematica

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 *)

PARI

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

Formula

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