A373086 - OEIS

%I #29 Jul 04 2024 09:21:42

%S 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 3130,9031,16807,0,1,127,2052,13686,54062,144495,262144,0,1,255,6297,

%U 57867,301321,1059261,2685817,4782969,0,1,511,19162,240049,1616764,7196785,23343742,56953279,100000000

%N 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 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 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 T(n, k) = k^n - Sum_{i=0..k-1} binomial(n, i)*(i+1)^(i-1)*(k-i-1)^(n-i).

%F T(n, k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n, i)*(i+1)^(i-1)*(i-k+1)^(n-i).

%e Table begins:

%e   1;

%e   0, 1;

%e   0, 1,   3;

%e   0, 1,   7,    16;

%e   0, 1,  15,    61,    125;

%e   0, 1,  31,   206,    671,    1296;

%e   0, 1,  63,   659,   3130,    9031,   16807;

%e   0, 1, 127,  2052,  13686,   54062,  144495,   262144;

%e   0, 1, 255,  6297,  57867,  301321, 1059261,  2685817,  4782969;

%e   0, 1, 511, 19162, 240049, 1616764, 7196785, 23343742, 56953279, 100000000;

%e   ...

%Y Cf. A000272 (diagonal), A000225 (column 2), A355645 (column 3, with offset 3).

%Y Cf. A260693 (first differences of columns).

%K nonn,tabl

%O 0,6

%A _Jayden Thadani_ and _Alan Kappler_, May 23 2024