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

0,6

Links

Table of n, a(n) for n=0..65.

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.

Example

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, ...
  ...

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

Crossrefs

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

Main diagonal gives A000272(n+1).

Cf. A264902.

Sequence in context: A162206 A075248 A359140 * A336707 A128325 A307883

Adjacent sequences: A365620 A365621 A365622 * A365624 A365625 A365626

Keyword

nonn,tabl

Author

J. Carlos Martínez Mori, Sep 13 2023

Status

approved