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A335088 a(n) = n! * [x^n] exp(Sum_{k=1..n, gcd(n,k) = 1} x^k / k).

%I A335088 #26 Oct 14 2020 15:13:31
%S A335088 1,1,1,4,9,96,145,4320,11025,179200,693441,36288000,62610625,
%T A335088 5748019200,15259154625,378002200576,4108830350625,334764638208000,
%U A335088 643373556450625,115242726703104000,309281197713168681,14870534508544000000,168541849294187630625
%N A335088 a(n) = n! * [x^n] exp(Sum_{k=1..n, gcd(n,k) = 1} x^k / k).
%C A335088 Number of permutations of [n] whose cycle lengths are relatively prime to n.
%H A335088 Alois P. Heinz, <a href="/A335088/b335088.txt">Table of n, a(n) for n = 0..450</a>
%p A335088 b:= proc(n, m) option remember; `if`(n=0, 1, add(`if`(
%p A335088       igcd(j, m)=1, b(n-j, m), 0)*(n-1)!/(n-j)!, j=1..n))
%p A335088     end:
%p A335088 a:= n-> b(n$2):
%p A335088 seq(a(n), n=0..23);  # _Alois P. Heinz_, Oct 12 2020
%t A335088 Table[n! SeriesCoefficient[Exp[Sum[Boole[GCD[n, k] == 1] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 22}]
%Y A335088 Cf. A074759, A226388, A335797.
%K A335088 nonn
%O A335088 0,4
%A A335088 _Ilya Gutkovskiy_, Oct 12 2020