A372842 a(n) is the number of parking functions of order n for which the second spot is lucky.

2, 11, 87, 908, 11824, 184944, 3381341, 70805696, 1671605646, 43938023168, 1272792377875, 40291409169408, 1383927524621468, 51265193822056448, 2037343816037147001, 86467962304018300928, 3903480077867017448410, 186771397981175865606144, 9441767566333191196904591

Offset: 2

Author

Kimberly P. Hadaway, Jun 20 2024

Keywords

• nonn

Comments

This sequence enumerates parking functions with lucky second spot (where a lucky spot is one which is parked in by a car which prefers that spot).

Examples

			For clarity, we write parentheses around parking functions. For n = 2, the a(2) = 2 solutions are the parking functions of length 2 with a lucky second spot are (1,2) and (2,1). The parking function (1,1) is not one of the solutions because the car which parks in the second spot did not prefer the second spot; this parking function does not contribute to our count.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..387
• Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 10.

Crossrefs

Cf. A000272 (for first spot), A372843 (for third spot), A372844 (for fourth spot), and A372845 (for fifth spot).

Programs

Mathematica

Array[(3/4)*(# + 1)^(# - 1) - (1/4)*(# - 1)^(# - 1) &, 19, 2] (* Michael De Vlieger, Jun 26 2024 *)

Python

def A372842(n): return 3*(n+1)**(n-1)-(n-1)**(n-1)>>2 # Chai Wah Wu, Jun 26 2024

Formula

a(n) = (3/4)*(n+1)^(n-1) - (1/4)*(n-1)^(n-1).