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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.
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%I #50 Sep 20 2023 18:19:16

%S 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,

%T 1,1,3,16,125,918,4683,5040,1,1,3,16,125,1171,9277,47293,40320,1,1,3,

%U 16,125,1296,12965,109438,545835,362880,1,1,3,16,125,1296,15511,166836,1475691,7087261,3628800

%N 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.

%F 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.

%e Square array T(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 3, 3, 3, 3, 3, 3, ...

%e 6, 13, 16, 16, 16, 16, 16, ...

%e 24, 75, 109, 125, 125, 125, 125, ...

%e 120, 541, 918, 1171, 1296, 1296, 1296, ...

%e 720, 4683, 9277, 12965, 15511, 16807, 16807, ...

%e ...

%p T:= proc(n, k) option remember; `if`(n=0, 1, add(min(i+1, k+1)*

%p binomial(n-1, i)*T(i, k)*T(n-1-i, k), i=0..n-1))

%p end:

%p seq(seq(T(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Sep 13 2023

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

%Y Main diagonal gives A000272(n+1).

%Y Cf. A264902.

%K nonn,tabl

%O 0,6

%A _J. Carlos Martínez Mori_, Sep 13 2023