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A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k. 4

%I

%S 1,2,1,8,5,3,50,34,25,16,432,307,243,189,125,4802,3506,2881,2401,1921,

%T 1296,65536,48729,40953,35328,30208,24583,16807,1062882,800738,683089,

%U 601441,531441,461441,379793,262144,20000000,15217031,13119879,11708091,10546875,9453125,8291909,6880121,4782969

%N Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.

%H D. Foata and J. Riordan, <a href="https://doi.org/10.1007/BF01834776">Mappings of acyclic and parking functions</a>, J. Aeq. Math., 10 (1974) 10-22.

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

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

%F T(n,k) = Sum_{j=k..n} A298594(n,j).

%F T(n,k) = (Sum_{j=k..n} A298597(n,j))/n.

%F Sum_{k=1..n} T(n,k) = A000272(n+1).

%e Triangle begins:

%e 1;

%e 2, 1;

%e 8, 5, 3;

%e 50, 34, 25, 16;

%e 432, 307, 243, 189, 125;

%e 4802, 3506, 2881, 2401, 1921, 1296;

%e 65536, 48729, 40953, 35328, 30208, 24583, 16807;

%e 1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144;

%e ...

%t Table[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. A000272, A298593, A298594, A298597.

%K easy,nonn,tabl

%O 1,2

%A _Rui Duarte_, Jan 22 2018

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Last modified July 28 23:26 EDT 2021. Contains 346340 sequences. (Running on oeis4.)