A375616 a(n) is the number of lucky cars in all parking functions of order n.

0, 1, 5, 36, 350, 4320, 64827, 1146880, 23383404, 540000000, 13933327265, 397303087104, 12407264266410, 421154777645056, 15439814208984375, 607985949695016960, 25593429637028941208, 1146928904801167933440, 54515427164280400691709, 2739404800000000000000000

Offset

0,3

Comments

This sequence enumerates lucky cars in parking functions of order n (where a lucky spot is one which is parked in by a car which prefers that spot).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..386

Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 10.

Formula

a(n) = Sum_{k=1..n} k*A370832(n,k) = Sum_{k=1..n} A374756(n,k).

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      expand(x*mul((n+1-k)+k*x, k=2..n)))
    end:
a:= n-> add(k*coeff(b(n), x, k), k=1..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 21 2024

Mathematica

b[n_] := b[n] = If[n == 0, 1, Expand[x*Product[(n+1-k) + k*x, {k, 2, n}]]];
a[n_] := Sum[k*Coefficient[b[n], x, k], {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 31 2024, after Alois P. Heinz *)

Crossrefs

Row sums of A374756.

Cf. A370832.

Sequence in context: A109186 A099391 A008785 * A355494 A081918 A062024

Adjacent sequences: A375613 A375614 A375615 * A375617 A375618 A375619

Keyword

nonn

Author

Kimberly P. Hadaway, Aug 21 2024, suggested by Andrew Howroyd

Extensions

a(6)-a(19) from Alois P. Heinz, Aug 21 2024

Status

approved