cp's OEIS Frontend

A370832 Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 0 <= k <= n.

%I A370832 #47 Jun 27 2024 08:47:54
%S A370832 1,0,1,0,1,2,0,2,8,6,0,6,37,58,24,0,24,204,504,444,120,0,120,1318,
%T A370832 4553,6388,3708,720,0,720,9792,44176,87296,81136,33984,5040,0,5040,
%U A370832 82332,463860,1203921,1582236,1064124,341136,40320,0,40320,773280,5270480,17164320,29724000,28328480,14602320,3733920,362880
%N A370832 Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 0 <= k <= n.
%C A370832 A car is called "lucky" if it gets its preferred parking spot.
%C A370832 Closely related to A220884.
%H A370832 Alois P. Heinz, <a href="/A370832/b370832.txt">Rows n = 0..140, flattened</a>
%H A370832 Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, <a href="https://arxiv.org/abs/2211.00536">Probabilistic Parking Functions</a>, arXiv:2211.00536 [math.CO], 2022.
%H A370832 FindStat, <a href="https://www.findstat.org/StatisticsDatabase/St000135/">St000135: The number of lucky cars of the parking function</a>.
%F A370832 T(n, n) = n!.
%F A370832 T(n, 1) = (n-1)!.
%F A370832 Sum_{k=1..n} T(n, k) = (n+1)^(n-1).
%F A370832 T(n+1, n) = A002538(n).
%F A370832 G.f. for row n>0: x * Product_{j=2..n} (n + 1 + j*(x-1)).
%F A370832 T(n, k) = [x^k] (x*(x - 1)^n*Pochhammer((n + x) / (x - 1), n)) / (n + x). - _Peter Luschny_, Jun 27 2024
%e A370832 Table begins:
%e A370832 n\k|  0     1     2      3       4       5       6      7     8
%e A370832 ---+-------------------------------------------------------------
%e A370832  0 |  1
%e A370832  1 |  0     1
%e A370832  2 |  0     1     2
%e A370832  3 |  0     2     8      6
%e A370832  4 |  0     6    37     58      24
%e A370832  5 |  0    24   204    504     444     120
%e A370832  6 |  0   120  1318   4553    6388    3708     720
%e A370832  7 |  0   720  9792  44176   87296   81136   33984   5040
%e A370832  8 |  0  5040 82332 463860 1203921 1582236 1064124 341136 40320
%e A370832       ...
%p A370832 b:= proc(n) option remember; `if`(n=0, 1,
%p A370832       expand(x*mul((n+1-k)+k*x, k=2..n)))
%p A370832     end:
%p A370832 T:= (n, k)-> coeff(b(n), x, k):
%p A370832 seq(seq(T(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jun 26 2024
%t A370832 row[n_] := (x (x - 1)^n Pochhammer[(n + x) / (x - 1), n]) / (n + x);
%t A370832 Table[CoefficientList[Series[row[n], {x, 0, n}], x], {n, 0, 8}] // Flatten
%t A370832 (* _Peter Luschny_, Jun 27 2024 *)
%Y A370832 Cf. A002538, A067948, A220884.
%Y A370832 Row sums give A000272(n+1).
%Y A370832 Cf. A000142 (main diagonal and column k=1 shifted).
%K A370832 nonn,tabl
%O A370832 0,6
%A A370832 _Peter Kagey_, Mar 02 2024
%E A370832 Edited by _Alois P. Heinz_, Jun 26 2024