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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. 17
1, 1, 3, 1, 16, 10, 1, 125, 107, 23, 1, 1296, 1346, 436, 46, 1, 16807, 19917, 8402, 1442, 87, 1, 262144, 341986, 173860, 41070, 4320, 162, 1, 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1, 100000000, 148717762, 96920092, 34268902, 6768184, 710314, 34660, 574, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008
FORMULA
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.
Sum_{k>0} k * T(n,k) = A036276(n-1) for n>0.
Sum_{k>0} T(n,k) = A101334(n).
Sum_{k>=0} (-1)^k * T(n,k) = A274279(n) for n>=1.
EXAMPLE
T(2,0) = 3: [1,1], [1,2], [2,1].
T(2,1) = 1: [2,2].
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].
T(3,2) = 1: [3,3,3].
Triangle T(n,k) begins:
0 : 1;
1 : 1;
2 : 3, 1;
3 : 16, 10, 1;
4 : 125, 107, 23, 1;
5 : 1296, 1346, 436, 46, 1;
6 : 16807, 19917, 8402, 1442, 87, 1;
7 : 262144, 341986, 173860, 41070, 4320, 162, 1;
8 : 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1;
...
MAPLE
S:= (n, k)-> `if`(k=0, n^n, add(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):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10);
MATHEMATICA
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 *)
CROSSREFS
Row sums give A000312.
T(2n,n) gives A264903.
Sequence in context: A071211 A222029 A038675 * A350446 A156653 A048159
KEYWORD
nonn,tabf,easy
AUTHOR
Alois P. Heinz, Nov 28 2015
STATUS
approved

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Last modified June 16 11:09 EDT 2024. Contains 373429 sequences. (Running on oeis4.)