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%I #31 Jul 05 2021 09:52:12
%S 1,1,2,3,16,25,396,721,11264,46089,602200,3628801,133494912,479001601,
%T 7692266960,95904273375,1914926104576,20922789888001,628693317946656,
%U 6402373705728001,182635841123840000,2496321046987530021,55826951075231672512,1124000727777607680001
%N Number of degree-n permutations of order dividing n. Number of solutions to x^n = 1 in S_n.
%H Alois P. Heinz, <a href="/A074759/b074759.txt">Table of n, a(n) for n = 0..450</a>
%F a(n) = n! * [x^n] exp(Sum_{k divides n} x^k/k).
%F a(n) = Sum_{d|n} A057731(n,d) for n >= 1. - _Alois P. Heinz_, Jul 05 2021
%p A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p add(mul(n-i, i=1..j-1)*A(n-j,k), j=numtheory[divisors](k))))
%p end:
%p a:= n-> A(n, n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 14 2013
%t Table[a = Sum[x^i/i, {i, Divisors[n]}]; Part[Range[0, 20]! CoefficientList[Series[Exp[a], {x, 0, 20}], x],n + 1], {n, 0, 20}] (* _Geoffrey Critzer_, Dec 04 2011 *)
%Y Cf. A057731, A074351, A261431.
%Y Main diagonal of A008307.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Sep 28 2002
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