A291132 - OEIS

%I #12 Feb 24 2019 08:41:53

%S 1,303,34660,2743112,181875244,11023248678,639875755364,

%T 36555471741284,2090131479753756,120898503338385149,

%U 7124218746544184628,429662666436736162636,26601747798152634836236,1694092238645618305809580,111106187207006959809867012

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

%H Alois P. Heinz, <a href="/A291132/b291132.txt">Table of n, a(n) for n = 7..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) ~ (43*exp(1)/720 - 88*exp(2)/15 + 405*exp(3)/8 - 368*exp(4)/3 + 235*exp(5)/2 - 48*exp(6) + 7*exp(7)) * 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, 6)-S(n, 7):

%p seq(a(n), n=7..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, 6] - S[n, 7];

%t Table[a[n], {n, 7, 23}] (* _Jean-François Alcover_, Feb 24 2019, from Maple *)

%Y Column k=6 of A264902.

%K nonn

%O 7,2

%A _Alois P. Heinz_, Aug 18 2017