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A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.

%I A281485 #33 Jan 31 2017 14:26:05
%S A281485 1,1,2,4,6,6,27,38,36,24,256,350,330,240,120,3125,4202,3960,3000,1800,
%T A281485 720,46656,62062,58506,45360,29400,15120,5040,823543,1087214,1025388,
%U A281485 806904,546000,312480,141120,40320,16777216,22024830,20781690,16524144,11493720,6985440,3598560,1451520,362880
%N A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.
%C A281485 A parking function of size n is a sequence (a_1,...,a_n) of positive integers such that, if b_1 <= b_2 <= ... <= b_n is the increasing rearrangement of the sequence (a_1,..,a_n), then b_i <= i.
%C A281485 Given a:[n]->[n], the center of a is the largest subset Z(a) = { z_1, ..., z_k } of [n] such that z_1 < z_2 < ... < z_k and a_(z_j) <= j, for every j in [k]. The length of the center of a is |Z(a)|.
%C A281485 Then T(n,k)= number of parking functions of size n with center of length k.
%H A281485 Rui Duarte and António Guedes de Oliveira, <a href="https://arxiv.org/abs/1611.03707">The number of parking functions with center of a given length</a>, arXiv:1611.03707 (2016).
%F A281485 T(n,k) = k*Sum_{j=0..k-1} (-1)^j*binomial(k-1,j)*(n-1-j)^(n-1).
%F A281485 T(n,k) = k!*Sum_{j_1+j_2+...+j_k=n-k} (n-1)^(j_1)*(n-2)^(j_2)*...*(n-k)^(j_k).
%e A281485 First seven rows:
%e A281485       1
%e A281485       1      2
%e A281485       4      6      6
%e A281485      27     38     36     24
%e A281485     256    350    330    240    120
%e A281485    3125   4202   3960   3000   1800    720
%e A281485   46656  62062  58506  45360  29400  15120   5040
%t A281485 Table[Which[n == k == 1, 1, k == 1, (n - 1)^(n - 1), k == n, n!, True, k Sum[(-1)^j*Binomial[k - 1, j] (n - 1 - j)^(n - 1), {j, 0, k - 1}]], {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Jan 23 2017 *)
%Y A281485 T(n,k) = k * A174551(n-1,k-1).
%Y A281485 T(n,1) = (n-1)^(n-1) = A000312(n-1).
%Y A281485 T(n,n-1) = n!(n-1)/2 = A001286(n), n>=2.
%Y A281485 T(n,n) = n! = A000142(n).
%Y A281485 Sum_{i=1,...,n} T(n,i) = (n+1)^(n-1) = A000272(n+1).
%K A281485 nonn,tabl
%O A281485 1,3
%A A281485 _Rui Duarte_, Jan 22 2017