A264903 Number of defective parking functions of length 2n and defect n.

1, 1, 23, 1442, 176843, 36046214, 11023248678, 4719570364004, 2693983725947891, 1976997422623843358, 1813499364725872444178, 2033181299894696684493980, 2735368952738645928181452734, 4349180440965667221581315433212, 8067655677482008559181766540571948

Offset: 0

Author

Alois P. Heinz, Nov 28 2015

Keywords

• nonn

Examples

			a(2) = 23: [1,4,4,4], [2,4,4,4], [3,3,3,3], [3,3,3,4], [3,3,4,3], [3,3,4,4], [3,4,3,3], [3,4,3,4], [3,4,4,3], [3,4,4,4], [4,1,4,4], [4,2,4,4], [4,3,3,3], [4,3,3,4], [4,3,4,3], [4,3,4,4], [4,4,1,4], [4,4,2,4], [4,4,3,3], [4,4,3,4], [4,4,4,1], [4,4,4,2], [4,4,4,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008

Crossrefs

Cf. A264902.

Programs

Maple

S:= (n, k)-> `if`(k=0, n^n, add(binomial(n, i)*k*
             (k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k)):
a:= n-> S(2*n, n)-S(2*n, n+1):
seq(a(n), n=0..20);

Mathematica

s[n_, k_] :=  Sum[Binomial[n, i]*k*(k + i)^(i - 1)*(n - k - i)^(n - i), {i, 0, n - k}]; Flatten[{1, Table[s[2*n, n] - s[2*n, n + 1], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 19 2017 *)
(* constant d *) 4*((1 - r)/(2 - r))^(2 - r)*((1 + r)/r)^r /.FindRoot[(((2 - r)*(1 + r))/((1 - r)*r))^(1 - r^2) == E^2, {r, 1/2}, WorkingPrecision -> 100] (* Vaclav Kotesovec, Aug 19 2017 *)

Formula

a(n) = A264902(2n,n).

a(n) ~ c * d^n * n^(2*n), where d = 4 * ((1-r)/(2-r))^(2-r) * ((1+r)/r)^r = 1.37946886318881879639758089832698445354075122787883765455607405487162077... where r = 0.3507604755943619981673674677676002458987390260372260977704596... is the root of the equation (((2-r)*(1+r))/((1-r)*r))^(1-r^2) = exp(2) and c = 0.71338164469811281152311896105657925861924201644973836628479626510877... - Vaclav Kotesovec, Aug 19 2017