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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.

%I A365623 #50 Sep 20 2023 18:19:16
%S A365623 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 A365623 1,1,3,16,125,918,4683,5040,1,1,3,16,125,1171,9277,47293,40320,1,1,3,
%U A365623 16,125,1296,12965,109438,545835,362880,1,1,3,16,125,1296,15511,166836,1475691,7087261,3628800
%N 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.
%F A365623 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 A365623 Square array T(n,k) begins:
%e A365623     1,    1,    1,     1,     1,     1,     1, ...
%e A365623     1,    1,    1,     1,     1,     1,     1, ...
%e A365623     2,    3,    3,     3,     3,     3,     3, ...
%e A365623     6,   13,   16,    16,    16,    16,    16, ...
%e A365623    24,   75,  109,   125,   125,   125,   125, ...
%e A365623   120,  541,  918,  1171,  1296,  1296,  1296, ...
%e A365623   720, 4683, 9277, 12965, 15511, 16807, 16807, ...
%e A365623   ...
%p A365623 T:= proc(n, k) option remember; `if`(n=0, 1, add(min(i+1, k+1)*
%p A365623        binomial(n-1, i)*T(i, k)*T(n-1-i, k), i=0..n-1))
%p A365623     end:
%p A365623 seq(seq(T(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Sep 13 2023
%Y A365623 Columns k=0..1, 3..4 give: A000142, A000670, A365626, A365627.
%Y A365623 Main diagonal gives A000272(n+1).
%Y A365623 Cf. A264902.
%K A365623 nonn,tabl
%O A365623 0,6
%A A365623 _J. Carlos Martínez Mori_, Sep 13 2023