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A298594 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.

%I A298594 #31 Dec 27 2024 08:46:51
%S A298594 1,1,1,3,2,3,16,9,9,16,125,64,54,64,125,1296,625,480,480,625,1296,
%T A298594 16807,7776,5625,5120,5625,7776,16807,262144,117649,81648,70000,70000,
%U A298594 81648,117649,262144,4782969,2097152,1411788,1161216,1093750,1161216,1411788,2097152,4782969
%N A298594 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.
%H A298594 Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See p. 6.
%F A298594 T(n,k) = binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).
%F A298594 T(n,k) = A298592(n,k) - A298592(n,k+1).
%F A298594 T(n,k) = (A298593(n,k) - A298593(n,k+1))/n.
%F A298594 T(n,k) = A298597(n,k)/n.
%F A298594 T(n,1) = A000272(n+2).
%F A298594 T(n,n) = A000272(n+2).
%F A298594 T(n,k) = T(n,n-k).
%e A298594 Triangle begins:
%e A298594        1;
%e A298594        1,      1;
%e A298594        3,      2,     3;
%e A298594       16,      9,     9,    16;
%e A298594      125,     64,    54,    64,   125;
%e A298594     1296,    625,   480,   480,   625,  1296;
%e A298594    16807,   7776,  5625,  5120,  5625,  7776,  16807;
%e A298594   262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144;
%e A298594   ...
%t A298594 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 A298594 Cf. A000272, A298592, A298593, A298597.
%K A298594 easy,nonn,tabl
%O A298594 1,4
%A A298594 _Rui Duarte_, Jan 22 2018