A291129 Number of defective parking functions of length n and defect three.

1, 46, 1442, 41070, 1166083, 34268902, 1059688652, 34723442062, 1208687001381, 44701813604150, 1754724115372438, 72987949807322222, 3210789166789472775, 149073947870611952326, 7289995971215959818304, 374726414201304528649678, 20207920615298454956444905

Offset: 4

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..387
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Crossrefs

Column k=3 of A264902.

Programs

Maple

S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
a:= n-> S(n, 3)-S(n, 4):
seq(a(n), n=4..23);

Mathematica

S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
a[n_] := S[n, 3] - S[n, 4];
Table[a[n], {n, 4, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Formula

a(n) ~ (-13*exp(1)/6 + 14*exp(2) - 15*exp(3) + 4*exp(4)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017