A365623 T(n,k) is the number of parking functions of length n with cars parking at most k spots away from their preferred spot; square array T(n,k), n>=0, k>=0, read by downward antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 3, 13, 24, 1, 1, 3, 16, 75, 120, 1, 1, 3, 16, 109, 541, 720, 1, 1, 3, 16, 125, 918, 4683, 5040, 1, 1, 3, 16, 125, 1171, 9277, 47293, 40320, 1, 1, 3, 16, 125, 1296, 12965, 109438, 545835, 362880, 1, 1, 3, 16, 125, 1296, 15511, 166836, 1475691, 7087261, 3628800

Offset: 0

Author

J. Carlos Martínez Mori, Sep 13 2023

Keywords

• nonn
• tabl

Examples

			Square array T(n,k) begins:
    1,    1,    1,     1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1,     1, ...
    2,    3,    3,     3,     3,     3,     3, ...
    6,   13,   16,    16,    16,    16,    16, ...
   24,   75,  109,   125,   125,   125,   125, ...
  120,  541,  918,  1171,  1296,  1296,  1296, ...
  720, 4683, 9277, 12965, 15511, 16807, 16807, ...
  ...
		

Crossrefs

Columns k=0..1, 3..4 give: A000142, A000670, A365626, A365627.

Main diagonal gives A000272(n+1).

Cf. A264902.

Programs

Maple

T:= proc(n, k) option remember; `if`(n=0, 1, add(min(i+1, k+1)*
       binomial(n-1, i)*T(i, k)*T(n-1-i, k), i=0..n-1))
    end:
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 13 2023

Formula

T(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * min(i+1,k+1) * T(i,k) * T(n-1-i,k) for n>0, T(0,k) = 1.