OFFSET
0,3
COMMENTS
Also the number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2^n.
Also the number of fixed-point free involutory automorphisms of the complete binary tree of height n.
FORMULA
a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 0.
a(n) ~ C^(2^n) for C = 1.467067423065535412629251121186749718727038915553188083467...
EXAMPLE
For n=2, the a(2)=3 fixed-point free involutions in C_2 wr C_2 (which is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).
MATHEMATICA
Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {0}, 9] (* Michael De Vlieger, Feb 25 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Nick Krempel, Feb 22 2020
STATUS
approved