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%I #42 Mar 05 2018 04:19:43
%S 1,4,2,24,15,9,200,136,100,64,2160,1535,1215,945,625,28812,21036,
%T 17286,14406,11526,7776,458752,341103,286671,247296,211456,172081,
%U 117649,8503056,6405904,5464712,4811528,4251528,3691528,3038344,2097152,180000000,136953279,118078911,105372819,94921875,85078125,74627181,61921089,43046721
%N Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.
%C T(n,k) is the number of pairs (f,i) such that f is a parking function and f(i) = k.
%F T(n,k) = n*Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).
%F T(n,k) = n*A298592(n,k).
%F T(n,k) = n*Sum_{j=k..n} A298594(n,j).
%F T(n,k) = Sum_{j=k..n} A298597(n,j).
%F Sum_{k=1..n} T(n,k) = n*A000272(n+1).
%F T(n+1,1) = A089946(n), T(n,n) = A000169(n). - _Andrey Zabolotskiy_, Feb 21 2018
%e Triangle begins:
%e ====================================================================
%e n\k| 1 2 3 4 5 6 7 8
%e ---|----------------------------------------------------------------
%e 1 | 1
%e 2 | 4 2
%e 3 | 24 15 9
%e 4 | 200 136 100 64
%e 5 | 2160 1535 1215 945 625
%e 6 | 28812 21036 17286 14406 11526 7776
%e 7 | 458752 341103 286671 247296 211456 172081 117649
%e 8 | 8503056 6405904 5464712 4811528 4251528 3691528 3038344 2097152
%e ...
%t 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 *)
%Y Cf. A000169, A000272, A089946, A298592, A298594, A298597.
%K easy,nonn,tabl
%O 1,2
%A _Rui Duarte_, Jan 22 2018
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