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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.
3

%I #22 Feb 20 2018 11:54:13

%S 1,2,2,9,6,9,64,36,36,64,625,320,270,320,625,7776,3750,2880,2880,3750,

%T 7776,117649,54432,39375,35840,39375,54432,117649,2097152,941192,

%U 653184,560000,560000,653184,941192,2097152,43046721,18874368,12706092,10450944,9843750,10450944,12706092,18874368,43046721

%N 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.

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

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

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

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

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

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

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

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

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

%e Triangle begins:

%e 1;

%e 2, 2;

%e 9, 6, 9;

%e 64, 36, 36, 64;

%e 625, 320, 270, 320, 625;

%e 7776, 3750, 2880, 2880, 3750, 7776;

%e 117649, 54432, 39375, 35840, 39375, 54432, 117649;

%e 2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152;

%e ...

%t 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 *)

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

%K easy,nonn,tabl

%O 1,2

%A _Rui Duarte_, Jan 22 2018