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A373086 Triangle read by rows: T(n, k) is the number of parking functions of length n with preferences restricted to {1, ..., k} for 0 <= k <= n.

%I A373086 #29 Jul 04 2024 09:21:42
%S A373086 1,0,1,0,1,3,0,1,7,16,0,1,15,61,125,0,1,31,206,671,1296,0,1,63,659,
%T A373086 3130,9031,16807,0,1,127,2052,13686,54062,144495,262144,0,1,255,6297,
%U A373086 57867,301321,1059261,2685817,4782969,0,1,511,19162,240049,1616764,7196785,23343742,56953279,100000000
%N A373086 Triangle read by rows: T(n, k) is the number of parking functions of length n with preferences restricted to {1, ..., k} for 0 <= k <= n.
%C A373086 Equivalently, this is the number of preference lists of n cars on k spots so that only n - k cars are unable to park.
%H A373086 P. J. Cameron, D. Johannsen, T. Prellberg, and P. Schweitzer, <a href="https://doi.org/10.37236/816">Counting Defective Parking Functions</a>, Electronic Journal of Combinatorics, Volume 15 (2008).
%F A373086 T(n, k) = k^n - Sum_{i=0..k-1} binomial(n, i)*(i+1)^(i-1)*(k-i-1)^(n-i).
%F A373086 T(n, k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n, i)*(i+1)^(i-1)*(i-k+1)^(n-i).
%e A373086 Table begins:
%e A373086   1;
%e A373086   0, 1;
%e A373086   0, 1,   3;
%e A373086   0, 1,   7,    16;
%e A373086   0, 1,  15,    61,    125;
%e A373086   0, 1,  31,   206,    671,    1296;
%e A373086   0, 1,  63,   659,   3130,    9031,   16807;
%e A373086   0, 1, 127,  2052,  13686,   54062,  144495,   262144;
%e A373086   0, 1, 255,  6297,  57867,  301321, 1059261,  2685817,  4782969;
%e A373086   0, 1, 511, 19162, 240049, 1616764, 7196785, 23343742, 56953279, 100000000;
%e A373086   ...
%Y A373086 Cf. A000272 (diagonal), A000225 (column 2), A355645 (column 3, with offset 3).
%Y A373086 Cf. A260693 (first differences of columns).
%K A373086 nonn,tabl
%O A373086 0,6
%A A373086 _Jayden Thadani_ and _Alan Kappler_, May 23 2024