A291134 - OEIS

%I #12 Feb 24 2019 08:42:10

%S 1,1103,269512,38643849,4218834608,393933602129,33499946915016,

%T 2693983725947891,209859823775671984,16093162912317174422,

%U 1228462028909579534968,94081283153407041089350,7269699339591280955315232,569088494101518607733459806

%N Number of defective parking functions of length n and defect eight.

%H Alois P. Heinz, <a href="/A291134/b291134.txt">Table of n, a(n) for n = 9..386</a>

%H Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, <a href="https://arxiv.org/abs/0803.0302">Counting Defective Parking Functions</a>, arXiv:0803.0302 [math.CO], 2008.

%F 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

%p S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):

%p a:= n-> S(n, 8)-S(n, 9):

%p seq(a(n), n=9..23);

%t S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];

%t a[n_] := S[n, 8] - S[n, 9];

%t Table[a[n], {n, 9, 23}] (* _Jean-François Alcover_, Feb 24 2019, from Maple *)

%Y Column k=8 of A264902.

%K nonn

%O 9,2

%A _Alois P. Heinz_, Aug 18 2017