A291135 Number of defective parking functions of length n and defect nine.

1, 2146, 754943, 143336610, 19795924787, 2267392009178, 231141766226605, 21881366451890002, 1976997422623843358, 173666031731576614842, 15025473411620865716938, 1292364106829281911023554, 111260031164008673095102874, 9635674549219284395173044506

Offset: 10

Author

Alois P. Heinz, Aug 18 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 10..386
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Crossrefs

Column k=9 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, 9)-S(n, 10):
seq(a(n), n=10..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, 9] - S[n, 10];
Table[a[n], {n, 10, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

Formula

a(n) ~ (-13*exp(1)/51840 + 92*exp(2)/315 - 7533*exp(3)/560 + 6016*exp(4)/45 - 11875*exp(5)/24 + 864*exp(6) - 4753*exp(7)/6 + 392*exp(8) - 99*exp(9) + 10*exp(10)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017