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

1, 574, 96620, 10358998, 886044810, 66943181150, 4719570364004, 320771944968342, 21454694483447459, 1431385710008667470, 96133394595460111056, 6540549310477955461846, 452777288307033641080180, 31990399760398854681388158, 2311790354938282481939931160

Offset: 8

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..386
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Crossrefs

Column k=7 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, 7)-S(n, 8):
seq(a(n), n=8..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, 7] - S[n, 8];
Table[a[n], {n, 8, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Formula

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