A372845 - OEIS

%I #11 Jul 09 2024 12:44:54

%S 625,8733,140298,2600879,54921875,1303885965,34409008596,999711522899,

%T 31719176377701,1091467041015625,40491113522829630,

%U 1611131116280526327,68448950529246552887,3092734133786108912869,148090628302001953125000,7491257174986774088059995,399205026805287676036911049

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

%C 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).

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

%e 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.

%t 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 *)

%o (Python)

%o 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

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

%K nonn

%O 5,1

%A _Kimberly P. Hadaway_, Jun 26 2024