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

0,3

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

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

Example

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].

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 *)

Crossrefs

Cf. A264902.

Sequence in context: A196422 A351508 A248703 * A003281 A330658 A034243

Adjacent sequences: A264900 A264901 A264902 * A264904 A264905 A264906

Keyword

nonn

Author

Alois P. Heinz, Nov 28 2015

Status

approved