A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.

1, 4, 2, 24, 15, 9, 200, 136, 100, 64, 2160, 1535, 1215, 945, 625, 28812, 21036, 17286, 14406, 11526, 7776, 458752, 341103, 286671, 247296, 211456, 172081, 117649, 8503056, 6405904, 5464712, 4811528, 4251528, 3691528, 3038344, 2097152, 180000000, 136953279, 118078911, 105372819, 94921875, 85078125, 74627181, 61921089, 43046721

Offset: 1

Author

Rui Duarte, Jan 22 2018

Keywords

• easy
• nonn
• tabl

Comments

T(n,k) is the number of pairs (f,i) such that f is a parking function and f(i) = k.

Examples

			Triangle begins:
====================================================================
n\k|       1       2       3       4       5       6       7       8
---|----------------------------------------------------------------
1  |       1
2  |       4       2
3  |      24      15       9
4  |     200     136     100      64
5  |    2160    1535    1215     945     625
6  |   28812   21036   17286   14406   11526    7776
7  |  458752  341103  286671  247296  211456  172081  117649
8  | 8503056 6405904 5464712 4811528 4251528 3691528 3038344 2097152
  ...
		

Crossrefs

Cf. A000169, A000272, A089946, A298592, A298594, A298597.

Programs

Mathematica

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

Formula

T(n,k) = n*Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).

T(n,k) = n*A298592(n,k).

T(n,k) = n*Sum_{j=k..n} A298594(n,j).

T(n,k) = Sum_{j=k..n} A298597(n,j).

Sum_{k=1..n} T(n,k) = n*A000272(n+1).

T(n+1,1) = A089946(n), T(n,n) = A000169(n). - Andrey Zabolotskiy, Feb 21 2018