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A264902 Number T(n,k) of defective parking functions of length n and defect k; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows.

%I A264902 #28 Aug 08 2022 20:42:22
%S A264902 1,1,3,1,16,10,1,125,107,23,1,1296,1346,436,46,1,16807,19917,8402,
%T A264902 1442,87,1,262144,341986,173860,41070,4320,162,1,4782969,6713975,
%U A264902 3924685,1166083,176843,12357,303,1,100000000,148717762,96920092,34268902,6768184,710314,34660,574,1
%N A264902 Number T(n,k) of defective parking functions of length n and defect k; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows.
%H A264902 Alois P. Heinz, <a href="/A264902/b264902.txt">Rows n = 0..141, flattened</a>
%H A264902 Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, <a href="https://arxiv.org/abs/0803.0302">Counting Defective Parking Functions</a>, arXiv:0803.0302 [math.CO], 2008
%F A264902 T(n,k) = S(n,k) - S(n,k+1) with S(n,0) = n^n, S(n,k) = Sum_{i=0..n-k} C(n,i) * k*(k+i)^(i-1) * (n-k-i)^(n-i) for k>0.
%F A264902 Sum_{k>0} k * T(n,k) = A036276(n-1) for n>0.
%F A264902 Sum_{k>0} T(n,k) = A101334(n).
%F A264902 Sum_{k>=0} (-1)^k * T(n,k) = A274279(n) for n>=1.
%e A264902 T(2,0) = 3: [1,1], [1,2], [2,1].
%e A264902 T(2,1) = 1: [2,2].
%e A264902 T(3,1) = 10: [1,3,3], [2,2,2], [2,2,3], [2,3,2], [2,3,3], [3,1,3], [3,2,2], [3,2,3], [3,3,1], [3,3,2].
%e A264902 T(3,2) = 1: [3,3,3].
%e A264902 Triangle T(n,k) begins:
%e A264902 0 :       1;
%e A264902 1 :       1;
%e A264902 2 :       3,       1;
%e A264902 3 :      16,      10,       1;
%e A264902 4 :     125,     107,      23,       1;
%e A264902 5 :    1296,    1346,     436,      46,      1;
%e A264902 6 :   16807,   19917,    8402,    1442,     87,     1;
%e A264902 7 :  262144,  341986,  173860,   41070,   4320,   162,   1;
%e A264902 8 : 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1;
%e A264902     ...
%p A264902 S:= (n, k)-> `if`(k=0, n^n, add(binomial(n, i)*k*
%p A264902             (k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k)):
%p A264902 T:= (n, k)-> S(n, k)-S(n, k+1):
%p A264902 seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10);
%t A264902 S[n_, k_] := If[k==0, n^n, Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}]]; T[n_, k_] := S[n, k]-S[n, k+1]; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, Max[0, n-1]}] // Flatten (* _Jean-François Alcover_, Feb 18 2017, translated from Maple *)
%Y A264902 Columns k=0-10 give: A000272(n+1), A140647, A291128, A291129, A291130, A291131, A291132, A291133, A291134, A291135, A291136.
%Y A264902 Row sums give A000312.
%Y A264902 T(2n,n) gives A264903.
%Y A264902 Cf. A036276, A101334, A274279.
%K A264902 nonn,tabf,easy
%O A264902 0,3
%A A264902 _Alois P. Heinz_, Nov 28 2015