A346086 Number of permutations of [2n] such that the GCD of the cycle lengths equals 2.

0, 1, 3, 105, 4725, 530145, 45270225, 12034447425, 2116670180625, 737902583042625, 219604524727012425, 137952599116097390625, 49583382753435146240625, 46991310794950147391390625, 25508895927267586991297765625, 24661803286201363305440202410625

Offset: 0

Author

Alois P. Heinz, Jul 04 2021

Keywords

• nonn

Examples

			a(1) = 1: (12).
a(2) = 3: (12)(34), (13)(24), (14)(23).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..225
• Wikipedia, Permutation

Crossrefs

Bisection of column k=2 of A346085 (even part).

Programs

Maple

b:= proc(n, g) option remember; `if`(n=0, `if`(g=2, 1, 0), `if`(g=1, 0,
       add(b(n-j, igcd(g, j))*binomial(n-1, j-1)*(j-1)!, j=2..n)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..19);

Mathematica

b[n_, g_] := b[n, g] = If[n == 0, If[g == 2, 1, 0], If[g == 1, 0,
     Sum[b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 2, n}]]];
a[n_] := b[2*n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

Formula

a(n) = A346085(2n,2).