A291131 Number of defective parking functions of length n and defect five.

1, 162, 12357, 710314, 36046214, 1735513330, 82324649310, 3930328083098, 191251911975191, 9558232936557458, 492897541966818651, 26298973648144245066, 1454100613639405907108, 83377530695619365120818, 4959049035365905488761452, 305903708967397161238879674

Offset: 6

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 6..386
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Crossrefs

Column k=5 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, 5)-S(n, 6):
seq(a(n), n=6..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, 5] - S[n, 6];
Table[a[n], {n, 6, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Formula

a(n) ~ (-31*exp(1)/120 + 32*exp(2)/3 - 99*exp(3)/2 + 68*exp(4) - 35*exp(5) + 6*exp(6)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017