A000782 a(n) = 2*Catalan(n) - Catalan(n-1).

1, 3, 8, 23, 70, 222, 726, 2431, 8294, 28730, 100776, 357238, 1277788, 4605980, 16715250, 61020495, 223931910, 825632610, 3056887680, 11360977650, 42368413620, 158498860260, 594636663660, 2236748680998, 8433988655580, 31872759742852, 120699748759856

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of Dyck (n+1)-paths that have a leading or trailing hill. - David Scambler, Aug 22 2012

a(n) is the number of parking functions of size n avoiding the patterns 132, 213, 312, and 321. - Lara Pudwell, Apr 10 2023

Number of Dyck (n+1)-paths that have exactly one return to the x-axis and/or a peak in the center of the path. - Roger Ford, May 15 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
• Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023).
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Anna Rodriguez Rasmussen, Exact Borel subalgebras of quasi-hereditary monomial algebras, arXiv:2504.01706 [math.RT], 2025. See p. 38.
• John R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349(4) (1997), 1285-1332.

Crossrefs

Partial sums of A071735.

Essentially the same as A061557.

Cf. A000108, A026012.

Programs

Magma

[2*Catalan(n)-Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012

Mathematica

CoefficientList[Series[(1+x*(1-(1-4*x)^(1/2))/(2*x)^1)*((1-(1-4*x)^(1/2))/(2*x))^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 10 2012 *)

Formula

Expansion of x*(1 + x*C)*C^2, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108.

Also, expansion of (1 + x^2*C^2)*C - 1, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108.

a(n) = (7*n - 5)/(n + 1) * C(n-1), where C(n) = A000108(n). - Ralf Stephan, Jan 13 2004

a(n) = leftmost column term of M^(n-1)*V, where M is a tridiagonal matrix with 1's in the super- and subdiagonals, (1, 2, 2, 2, ...) in the main diagonal, and the rest zeros; and V is the vector [1, 2, 0, 0, 0, ...]. - Gary W. Adamson, Jun 16 2011

a(n) = A000108(n+1) - A026012(n-1). - David Scambler, Aug 22 2012