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A370832
Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 0 <= k <= n.
3
1, 0, 1, 0, 1, 2, 0, 2, 8, 6, 0, 6, 37, 58, 24, 0, 24, 204, 504, 444, 120, 0, 120, 1318, 4553, 6388, 3708, 720, 0, 720, 9792, 44176, 87296, 81136, 33984, 5040, 0, 5040, 82332, 463860, 1203921, 1582236, 1064124, 341136, 40320, 0, 40320, 773280, 5270480, 17164320, 29724000, 28328480, 14602320, 3733920, 362880
OFFSET
0,6
COMMENTS
A car is called "lucky" if it gets its preferred parking spot.
Closely related to A220884.
LINKS
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
Row sums give A000272(n+1).
Cf. A000142 (main diagonal and column k=1 shifted).
Sequence in context: A301772 A021497 A201735 * A029593 A196504 A182550
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Mar 02 2024
EXTENSIONS
Edited by Alois P. Heinz, Jun 26 2024
STATUS
approved