A291130 Number of defective parking functions of length n and defect four.

1, 87, 4320, 176843, 6768184, 256059854, 9846223168, 390516805362, 16102219296008, 693122084961945, 31208245366326896, 1470819863019421317, 72549461960461640120, 3743176448672690767272, 201836660477563528892704, 11362223977488695430091444

Offset: 5

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..386
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Crossrefs

Column k=4 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, 4)-S(n, 5):
seq(a(n), n=5..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, 4] - S[n, 5];
Table[a[n], {n, 5, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Formula

a(n) ~ (7*exp(1)/8 - 44*exp(2)/3 + 69*exp(3)/2 - 24*exp(4) + 5*exp(5)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017