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A291133 Number of defective parking functions of length n and defect seven.

%I A291133 #12 Feb 24 2019 08:42:20
%S A291133 1,574,96620,10358998,886044810,66943181150,4719570364004,
%T A291133 320771944968342,21454694483447459,1431385710008667470,
%U A291133 96133394595460111056,6540549310477955461846,452777288307033641080180,31990399760398854681388158,2311790354938282481939931160
%N A291133 Number of defective parking functions of length n and defect seven.
%H A291133 Alois P. Heinz, <a href="/A291133/b291133.txt">Table of n, a(n) for n = 8..386</a>
%H A291133 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 A291133 a(n) ~ (-19*exp(1)/1680 + 116*exp(2)/45 - 1593*exp(3)/40 + 160*exp(4) - 1525*exp(5)/6 + 186*exp(6) - 63*exp(7) + 8*exp(8)) * n^(n-1). - _Vaclav Kotesovec_, Aug 19 2017
%p A291133 S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
%p A291133 a:= n-> S(n, 7)-S(n, 8):
%p A291133 seq(a(n), n=8..23);
%t A291133 S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
%t A291133 a[n_] := S[n, 7] - S[n, 8];
%t A291133 Table[a[n], {n, 8, 23}] (* _Jean-François Alcover_, Feb 24 2019, from Maple *)
%Y A291133 Column k=7 of A264902.
%K A291133 nonn
%O A291133 8,2
%A A291133 _Alois P. Heinz_, Aug 18 2017