A264902 - OEIS

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.

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

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OFFSET

0,3

LINKS

Alois P. Heinz, Rows n = 0..141, flattened

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

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

Row sums give A000312.

T(2n,n) gives A264903.

Cf. A036276, A101334, A274279.

Sequence in context: A071211 A222029 A038675 * A350446 A156653 A048159

Adjacent sequences: A264899 A264900 A264901 * A264903 A264904 A264905

KEYWORD

nonn,tabf,easy

AUTHOR

Alois P. Heinz, Nov 28 2015

STATUS

approved