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

%I A298592 #36 Dec 27 2024 08:48:17
%S A298592 1,2,1,8,5,3,50,34,25,16,432,307,243,189,125,4802,3506,2881,2401,1921,
%T A298592 1296,65536,48729,40953,35328,30208,24583,16807,1062882,800738,683089,
%U A298592 601441,531441,461441,379793,262144,20000000,15217031,13119879,11708091,10546875,9453125,8291909,6880121,4782969
%N A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.
%H A298592 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.
%H A298592 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 A298592 T(n,k) = Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).
%F A298592 T(n,k) = A298593(n,k)/n.
%F A298592 T(n,k) = Sum_{j=k..n} A298594(n,j).
%F A298592 T(n,k) = (Sum_{j=k..n} A298597(n,j))/n.
%F A298592 Sum_{k=1..n} T(n,k) = A000272(n+1).
%e A298592 Triangle begins:
%e A298592         1;
%e A298592         2,      1;
%e A298592         8,      5,      3;
%e A298592        50,     34,     25,     16;
%e A298592       432,    307,    243,    189,    125;
%e A298592      4802,   3506,   2881,   2401,   1921,   1296;
%e A298592     65536,  48729,  40953,  35328,  30208,  24583,  16807;
%e A298592   1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144;
%e A298592   ...
%t A298592 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 A298592 Cf. A000272, A298593, A298594, A298597.
%K A298592 easy,nonn,tabl
%O A298592 1,2
%A A298592 _Rui Duarte_, Jan 22 2018