OFFSET
1,2
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.
J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349(4) (1997), 1285-1332.
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
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 *)
PROG
(Magma) [2*Catalan(n)-Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved