A291128 Number of defective parking functions of length n and defect two.

1, 23, 436, 8402, 173860, 3924685, 96920092, 2612981360, 76612170196, 2432096760755, 83225580995116, 3056917610828590, 120045033150878404, 5021755110536666777, 223031850751250882620, 10484575680391970139980, 520227143451578652486196, 27175721567427682443046975

Offset: 3

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..387
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008

Crossrefs

Column k=2 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, 2)-S(n, 3):
seq(a(n), n=3..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, 2] - S[n, 3];
Table[a[n], {n, 3, 23}] (* Jean-François Alcover, Aug 20 2018, from Maple *)

Formula

a(n) ~ (7*exp(1)/2 - 8*exp(2) + 3*exp(3)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017