A298593 - OEIS

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

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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