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A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 A140331

Adjacent sequences:  A298590 A298591 A298592 * A298594 A298595 A298596

KEYWORD

easy,nonn,tabl

AUTHOR

Rui Duarte, Jan 22 2018

STATUS

approved

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Last modified November 17 19:02 EST 2018. Contains 317276 sequences. (Running on oeis4.)