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

%I A346086 #14 Mar 06 2022 04:44:13
%S A346086 0,1,3,105,4725,530145,45270225,12034447425,2116670180625,
%T A346086 737902583042625,219604524727012425,137952599116097390625,
%U A346086 49583382753435146240625,46991310794950147391390625,25508895927267586991297765625,24661803286201363305440202410625
%N A346086 Number of permutations of [2n] such that the GCD of the cycle lengths equals 2.
%H A346086 Alois P. Heinz, <a href="/A346086/b346086.txt">Table of n, a(n) for n = 0..225</a>
%H A346086 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A346086 a(n) = A346085(2n,2).
%e A346086 a(1) = 1: (12).
%e A346086 a(2) = 3: (12)(34), (13)(24), (14)(23).
%p A346086 b:= proc(n, g) option remember; `if`(n=0, `if`(g=2, 1, 0), `if`(g=1, 0,
%p A346086        add(b(n-j, igcd(g, j))*binomial(n-1, j-1)*(j-1)!, j=2..n)))
%p A346086     end:
%p A346086 a:= n-> b(2*n, 0):
%p A346086 seq(a(n), n=0..19);
%t A346086 b[n_, g_] := b[n, g] = If[n == 0, If[g == 2, 1, 0], If[g == 1, 0,
%t A346086      Sum[b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 2, n}]]];
%t A346086 a[n_] := b[2*n, 0];
%t A346086 Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Mar 06 2022, after _Alois P. Heinz_ *)
%Y A346086 Bisection of column k=2 of A346085 (even part).
%K A346086 nonn
%O A346086 0,3
%A A346086 _Alois P. Heinz_, Jul 04 2021