A298597 Number T(n,k) of times the value k appears on the parking functions of length n and such that if we replace that value k with k+1 we don't get a parking function.

1, 2, 2, 9, 6, 9, 64, 36, 36, 64, 625, 320, 270, 320, 625, 7776, 3750, 2880, 2880, 3750, 7776, 117649, 54432, 39375, 35840, 39375, 54432, 117649, 2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152, 43046721, 18874368, 12706092, 10450944, 9843750, 10450944, 12706092, 18874368, 43046721

Offset: 1

Author

Rui Duarte, Jan 22 2018

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
        1;
        2,      2;
        9,      6,      9;
       64,     36,     36,     64;
      625,    320,    270,    320,    625;
     7776,   3750,   2880,   2880,   3750,   7776;
   117649,  54432,  39375,  35840,  39375,  54432, 117649;
  2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152;
  ...
		

Crossrefs

Cf. A000169, A000272, A298592, A298593, A298594.

Programs

Mathematica

Table[n Binomial[n - 1, k - 1] k^(k - 2)*(n + 1 - k)^(n - 1 - k), {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

Formula

T(n,k) = n*binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).

T(n,k) = n*A298594(n,k).

T(n.k) = A298593(n,k)-A298593(n,k+1).

T(n,k) = n*(A298592(n,k)-A298592(n,k+1)).

T(n,1) = n*A000272(n+2).

T(n,n) = n*A000272(n+2).

T(n,1) = A000169(n).

T(n,n) = A000169(n).

T(n,k) = T(n,n-k).