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Triangle read by rows: T(n,k) = number of parking functions a of length n such that a(1) = k and if we replace a(1) = k with k+1 we don't get a parking function.
3

%I #27 Feb 20 2018 11:55:21

%S 1,1,1,3,2,3,16,9,9,16,125,64,54,64,125,1296,625,480,480,625,1296,

%T 16807,7776,5625,5120,5625,7776,16807,262144,117649,81648,70000,70000,

%U 81648,117649,262144,4782969,2097152,1411788,1161216,1093750,1161216,1411788,2097152,4782969

%N Triangle read by rows: T(n,k) = number of parking functions a of length n such that a(1) = k and if we replace a(1) = k with k+1 we don't get a parking function.

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

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

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

%F T(n,k) = A298597(n,k)/n.

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

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

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

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 2, 3;

%e 16, 9, 9, 16;

%e 125, 64, 54, 64, 125;

%e 1296, 625, 480, 480, 625, 1296;

%e 16807, 7776, 5625, 5120, 5625, 7776, 16807;

%e 262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144;

%e ...

%t Table[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. A000272, A298592, A298593, A298597.

%K easy,nonn,tabl

%O 1,4

%A _Rui Duarte_, Jan 22 2018