A386965 Number of permutations p of [2*n] such that there is at least one index i in [2*n-1] with p(i+1) = n + p(i).

1, 10, 294, 16296, 1458120, 191751120, 34807535280, 8337722440320, 2547572372311680, 966944845408147200, 446304490431888211200, 246166572372916851532800, 159902551429370021259187200, 120818209587660157360960972800, 105060730670227917425027835648000

Offset

1,2

Comments

Problem 6 at IMO '89 essentially asks to show that a(n) > (2*n)!/4.

Links

Table of n, a(n) for n=1..15.

30th International Mathematical Olympiad (1989), Problem 6.

Index to sequences related to Olympiads.

Formula

a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (2*n - k)!.

a(n) = (2*n)! * (1 - 1F1(-n; -2*n; -1)).

a(n) = n! * A324361(n).

Example

The 10 permutations corresponding to a(2) are 1243, 1324, 1342, 2134, 2413, 2431, 3124, 3241, 4132, 4213.

Mathematica

a[n_] := Sum[(-1)^(k+1) Binomial[n, k] (2 n - k)!, {k, n}]; Array[a, 15]

Crossrefs

Cf. A002674, A324361, A159960.

Sequence in context: A291658 A159960 A258794 * A239775 A059072 A000459

Adjacent sequences: A386962 A386963 A386964 * A386966 A386967 A386968

Keyword

nonn,easy

Author

Giovanni Resta, Aug 11 2025

Status

approved