A036276 -id:A036276 - cp's OEIS frontend

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

Offset: 0

Alois P. Heinz, Nov 28 2015

			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;
    ...

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

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.

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);

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 *)

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.

A368982 Triangle read by rows: T(n, k) = binomial(n, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k) / 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 9, 3, 0, 0, 96, 36, 24, 0, 0, 1250, 480, 360, 270, 0, 0, 19440, 7500, 5760, 4860, 3840, 0, 0, 352947, 136080, 105000, 90720, 80640, 65625, 0, 0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0

Offset: 0

Peter Luschny, Jan 11 2024

			Triangle starts:
  [0] [0]
  [1] [0,       0]
  [2] [0,       1,       0]
  [3] [0,       9,       3,       0]
  [4] [0,      96,      36,      24,       0]
  [5] [0,    1250,     480,     360,     270,       0]
  [6] [0,   19440,    7500,    5760,    4860,    3840,       0]
  [7] [0,  352947,  136080,  105000,   90720,   80640,   65625,       0]
  [8] [0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0]

A368849, A369016 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, k) = A368849(n, k) / 2.
T(n, 1) = A081131(n) for n >= 1.
T(n, n - 1) = A081133(n - 2) for n >= 2.
Sum_{k=0..n} T(n, k) = A036276(n - 1) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / 2 for n >= 0.
Cf. A369072, A369025.

T := (n, k) -> binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)/2:
seq(seq(T(n, k), k = 0..n), n=0..9);

A368982[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k)/2; Table[A368982[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)

def T(n, k): return binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)//2
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])

T = A369072 - A369025.

cp's OEIS Frontend

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.

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