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

1,2

Comments

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

Links

Table of n, a(n) for n=1..45.

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

Example

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
  ...

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 *)

Crossrefs

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

Sequence in context: A241437 A030211 A134461 * A228474 A058167 A372849

Adjacent sequences: A298590 A298591 A298592 * A298594 A298595 A298596

Keyword

easy,nonn,tabl

Author

Rui Duarte, Jan 22 2018

Status

approved