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A343576
Number of permutations of [n] without fixed points and all cycles equal length.
1
1, 0, 1, 2, 9, 24, 175, 720, 6405, 42560, 436401, 3628800, 48073795, 479001600, 7116730335, 88966701824, 1474541093025, 20922789888000, 400160588853025, 6402373705728000, 133991603578884051, 2457732174030848000, 55735573291977790575, 1124000727777607680000
OFFSET
0,4
LINKS
FORMULA
a(n) = Sum_{d|n, d<n} (n!/d!)*(d/n)^d for n>0, a(0) = 1.
a(n) = A261431(n) for n in { A000040, A001358 }.
EXAMPLE
a(4) = 9: (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (2,3,4,1), (2,4,1,3), (3,1,4,2), (3,4,2,1), (4,1,2,3), (4,3,1,2).
MAPLE
a:= n-> `if`(n=0, 1, add(n!/d!*(d/n)^d, d=numtheory[divisors](n) minus {n})):
seq(a(n), n=0..23); # Alois P. Heinz, Apr 20 2021
PROG
(PARI) a(n) = if (n, sumdiv(n, d, if (d<n, (n!/d!)*(d/n)^d)), 1); \\ Michel Marcus, Apr 21 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary Yane, Apr 20 2021
STATUS
approved