A372845 a(n) is the number of parking functions of order n for which the fifth spot is lucky.

625, 8733, 140298, 2600879, 54921875, 1303885965, 34409008596, 999711522899, 31719176377701, 1091467041015625, 40491113522829630, 1611131116280526327, 68448950529246552887, 3092734133786108912869, 148090628302001953125000, 7491257174986774088059995, 399205026805287676036911049

Offset

5,1

Comments

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

Links

Table of n, a(n) for n=5..21.

Formula

a(n) = (3/5)*(n+1)^(n-1) - (1/30)*(118*n^3 - 531*n^2 + 659*n - 192)*(n-4)^(n-4).

Example

For clarity, we write parentheses around parking functions. For n = 6, there are a(6) = 8733 solutions. An example of a parking function of order 6 with a lucky fifth spot is (1,4,1,5,2,2); here, the fourth car parks in the fifth spot which is its preferred spot. This parking function contributes to our count. A non-example is the parking function (1,1,1,1,1,5); here, the fifth car parks in the fifth spot, but its preference is spot 1. This parking function does not contribute to our count.

Mathematica

a[n_]:=(3/5)*(n+1)^(n-1)-(1/30)*(118*n^3-531*n^2+659*n-192)*(n-4)^(n-4); Array[a, 17, 5] (* Stefano Spezia, Jun 26 2024 *)

Prog

(Python)
def A372845(n): return (18*(n+1)**(n-1)-(n*(n*(59*((n<<1)-9))+659)-192)*(n-4)**(n-4))//30 # Chai Wah Wu, Jun 26 2024

Crossrefs

Cf. A000272 (for first spot), A372842 (for second spot), A372843 (for third spot), and A372844 (for fourth spot).

Sequence in context: A143026 A238700 A064781 * A250832 A250827 A055868

Adjacent sequences: A372842 A372843 A372844 * A372846 A372847 A372848

Keyword

nonn

Author

Kimberly P. Hadaway, Jun 26 2024

Status

approved