OFFSET
0,6
COMMENTS
A car is called "lucky" if it gets its preferred parking spot.
Closely related to A220884.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, Probabilistic Parking Functions, arXiv:2211.00536 [math.CO], 2022.
FORMULA
T(n, n) = n!.
T(n, 1) = (n-1)!.
Sum_{k=1..n} T(n, k) = (n+1)^(n-1).
T(n+1, n) = A002538(n).
G.f. for row n>0: x * Product_{j=2..n} (n + 1 + j*(x-1)).
T(n, k) = [x^k] (x*(x - 1)^n*Pochhammer((n + x) / (x - 1), n)) / (n + x). - Peter Luschny, Jun 27 2024
EXAMPLE
Table begins:
n\k| 0 1 2 3 4 5 6 7 8
---+-------------------------------------------------------------
0 | 1
1 | 0 1
2 | 0 1 2
3 | 0 2 8 6
4 | 0 6 37 58 24
5 | 0 24 204 504 444 120
6 | 0 120 1318 4553 6388 3708 720
7 | 0 720 9792 44176 87296 81136 33984 5040
8 | 0 5040 82332 463860 1203921 1582236 1064124 341136 40320
...
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
expand(x*mul((n+1-k)+k*x, k=2..n)))
end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Jun 26 2024
MATHEMATICA
row[n_] := (x (x - 1)^n Pochhammer[(n + x) / (x - 1), n]) / (n + x);
Table[CoefficientList[Series[row[n], {x, 0, n}], x], {n, 0, 8}] // Flatten
(* Peter Luschny, Jun 27 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Mar 02 2024
EXTENSIONS
Edited by Alois P. Heinz, Jun 26 2024
STATUS
approved