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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.
17

%I #28 Aug 08 2022 20:42:22

%S 1,1,3,1,16,10,1,125,107,23,1,1296,1346,436,46,1,16807,19917,8402,

%T 1442,87,1,262144,341986,173860,41070,4320,162,1,4782969,6713975,

%U 3924685,1166083,176843,12357,303,1,100000000,148717762,96920092,34268902,6768184,710314,34660,574,1

%N 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 Alois P. Heinz, <a href="/A264902/b264902.txt">Rows n = 0..141, flattened</a>

%H 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 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 Sum_{k>0} k * T(n,k) = A036276(n-1) for n>0.

%F Sum_{k>0} T(n,k) = A101334(n).

%F Sum_{k>=0} (-1)^k * T(n,k) = A274279(n) for n>=1.

%e T(2,0) = 3: [1,1], [1,2], [2,1].

%e T(2,1) = 1: [2,2].

%e 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 T(3,2) = 1: [3,3,3].

%e Triangle T(n,k) begins:

%e 0 : 1;

%e 1 : 1;

%e 2 : 3, 1;

%e 3 : 16, 10, 1;

%e 4 : 125, 107, 23, 1;

%e 5 : 1296, 1346, 436, 46, 1;

%e 6 : 16807, 19917, 8402, 1442, 87, 1;

%e 7 : 262144, 341986, 173860, 41070, 4320, 162, 1;

%e 8 : 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1;

%e ...

%p S:= (n, k)-> `if`(k=0, n^n, add(binomial(n, i)*k*

%p (k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k)):

%p T:= (n, k)-> S(n, k)-S(n, k+1):

%p seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10);

%t 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 Columns k=0-10 give: A000272(n+1), A140647, A291128, A291129, A291130, A291131, A291132, A291133, A291134, A291135, A291136.

%Y Row sums give A000312.

%Y T(2n,n) gives A264903.

%Y Cf. A036276, A101334, A274279.

%K nonn,tabf,easy

%O 0,3

%A _Alois P. Heinz_, Nov 28 2015