A291134 Number of defective parking functions of length n and defect eight.

1, 1103, 269512, 38643849, 4218834608, 393933602129, 33499946915016, 2693983725947891, 209859823775671984, 16093162912317174422, 1228462028909579534968, 94081283153407041089350, 7269699339591280955315232, 569088494101518607733459806

Offset: 9

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..386
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Crossrefs

Column k=8 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, 8)-S(n, 9):
seq(a(n), n=9..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, 8] - S[n, 9];
Table[a[n], {n, 9, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Formula

a(n) ~ (73*exp(1)/40320 - 296*exp(2)/315 + 405*exp(3)/16 - 2432*exp(4)/15 + 9625*exp(5)/24 - 468*exp(6) + 553*exp(7)/2 - 80*exp(8) + 9*exp(9)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017