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%I #15 Mar 27 2020 13:41:00
%S 1,3,8,50,144,1764,8448,89424,576000,10628640,57231360,1486442880,
%T 11285084160,196771680000,2643856588800,70734282393600,
%U 558255985459200,22376988058521600,227061389721600000,6244741918808064000,106778305830518784000,4148476779335454720000
%N The number of n-permutations having precisely two cycles whose lengths are relatively prime.
%C a(n) is the coefficient of x^n/n! in the Taylor series expansion of B(A(x)) where A(x)= Sum_{positive integers relatively prime to n}x^n/n and B(x)=x^2/2!.
%H Andrew Howroyd, <a href="/A194364/b194364.txt">Table of n, a(n) for n = 2..200</a>
%F a(p) = A000254(p-1) for prime p.
%t f[list_]:=x^First[list]/First[list]+x^Last[list]/Last[list];
%t Prepend[Table[a=Total[Map[f, Select[IntegerPartitions[n, 2], Apply[GCD, #]==1&]]]; Last[Range[0, n]! CoefficientList[Series[a^2/2!, {x, 0, n}], x]], {n, 3, 30}], 1]
%o (PARI) a(n)={sum(k=1, n-1, if(gcd(k, n-k)==1, binomial(n,k)*(k-1)!*(n-k-1)!))/2} \\ _Andrew Howroyd_, Mar 27 2020
%Y Cf. A000254.
%K nonn
%O 2,2
%A _Geoffrey Critzer_, Oct 12 2011
%E Terms a(19) and beyond from _Andrew Howroyd_, Mar 27 2020
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