A239840 Number of ordered pairs of permutation functions (f,g) on n elements satisfying f(x) = f(g(g(x))).

1, 1, 4, 24, 240, 3120, 54720, 1169280, 30804480, 950745600, 34459084800, 1424870092800, 67133032243200, 3540086232883200, 208397961547776000, 13533822947893248000, 966773828738285568000, 75334352557782269952000, 6385175803136642383872000

Offset: 0

Author

Chad Brewbaker, Mar 27 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Programs

Maple

a:= proc(n) a(n):= `if`(n<2, 1, n*a(n-1) +n*(n-1)^2*a(n-2)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2014

Mathematica

a[n_] := a[n] = n a[n-1] + n(n-1)^2 a[n-2]; a[0] = a[1] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Oct 04 2019 *)

Formula

From Alois P. Heinz, Jul 23 2014: (Start)

a(n) = n! * A000085(n) = A000142(n) * A000085(n).

a(n) = n*a(n-1) + n*(n-1)^2*a(n-2) for n>=2, a(0) = a(1) = 1. (End)

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x + x^2 / 2). - Ilya Gutkovskiy, Jul 15 2021

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014

a(10)-a(18) from Alois P. Heinz, Jul 23 2014