A194364 The number of n-permutations having precisely two cycles whose lengths are relatively prime.

1, 3, 8, 50, 144, 1764, 8448, 89424, 576000, 10628640, 57231360, 1486442880, 11285084160, 196771680000, 2643856588800, 70734282393600, 558255985459200, 22376988058521600, 227061389721600000, 6244741918808064000, 106778305830518784000, 4148476779335454720000

Offset

2,2

Comments

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!.

Links

Andrew Howroyd, Table of n, a(n) for n = 2..200

Formula

a(p) = A000254(p-1) for prime p.

Mathematica

f[list_]:=x^First[list]/First[list]+x^Last[list]/Last[list];
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]

Prog

(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

Crossrefs

Cf. A000254.

Sequence in context: A316797 A000862 A306042 * A338439 A005444 A222684

Adjacent sequences: A194361 A194362 A194363 * A194365 A194366 A194367

Keyword

nonn

Author

Geoffrey Critzer, Oct 12 2011

Extensions

Terms a(19) and beyond from Andrew Howroyd, Mar 27 2020

Status

approved