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

%I A239840 #23 Jul 15 2021 16:39:53
%S A239840 1,1,4,24,240,3120,54720,1169280,30804480,950745600,34459084800,
%T A239840 1424870092800,67133032243200,3540086232883200,208397961547776000,
%U A239840 13533822947893248000,966773828738285568000,75334352557782269952000,6385175803136642383872000
%N A239840 Number of ordered pairs of permutation functions (f,g) on n elements satisfying f(x) = f(g(g(x))).
%H A239840 Alois P. Heinz, <a href="/A239840/b239840.txt">Table of n, a(n) for n = 0..300</a>
%F A239840 From _Alois P. Heinz_, Jul 23 2014: (Start)
%F A239840 a(n) = n! * A000085(n) = A000142(n) * A000085(n).
%F A239840 a(n) = n*a(n-1) + n*(n-1)^2*a(n-2) for n>=2, a(0) = a(1) = 1. (End)
%F A239840 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x + x^2 / 2). - _Ilya Gutkovskiy_, Jul 15 2021
%p A239840 a:= proc(n) a(n):= `if`(n<2, 1, n*a(n-1) +n*(n-1)^2*a(n-2)) end:
%p A239840 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 23 2014
%t A239840 a[n_] := a[n] = n a[n-1] + n(n-1)^2 a[n-2]; a[0] = a[1] = 1;
%t A239840 a /@ Range[0, 20] (* _Jean-François Alcover_, Oct 04 2019 *)
%Y A239840 Cf. A000012, A000085, A000142, A088311, A053529, A001044.
%K A239840 nonn
%O A239840 0,3
%A A239840 _Chad Brewbaker_, Mar 27 2014
%E A239840 a(8)-a(9) from _Giovanni Resta_, Mar 27 2014
%E A239840 a(10)-a(18) from _Alois P. Heinz_, Jul 23 2014