A332759 - OEIS

%I #26 Feb 27 2020 21:54:21

%S 1,1,2,2,6,6,12,12,44,44,88,88,264,264,528,528,2064,2064,4128,4128,

%T 12384,12384,24768,24768,90816,90816,181632,181632,544896,544896,

%U 1089792,1089792,4292864,4292864,8585728,8585728,25757184,25757184,51514368,51514368

%N Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree n.

%C As the Sylow 2-subgroups of S_(2n) are isomorphic to those of S_(2n+1), the terms of this sequence come in pairs.

%C Also the number of involutory automorphisms (including identity) of the full binary tree with n leaves (hence 2n-1 vertices) in which all left children are complete (perfect) binary trees.

%F a(n) = Product(A332757(k)) where k ranges over the positions of 1 bits in the binary expansion of n.

%F a(n) = big-Theta(C^n) for C = 1.6116626399..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).

%F Conjecture: B=1 and A=0.409091077245262341747187571213565366725933766222357989... - _Vaclav Kotesovec_, Feb 26 2020

%e For n=4, the a(4)=6 elements satisfying x^2=1 in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).

%p b:= proc(n) b(n):=`if`(n=0, 1, b(n-1)^2+2^(2^(n-1)-1)) end:

%p a:= n-> (l-> mul(`if`(l[i]=1, b(i-1), 1), i=1..nops(l)))(Bits[Split](n)):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 27 2020

%t Join[{1}, Block[{nn = 33, s}, s = Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {1}, Ceiling@ Log2@ nn]; Array[Times @@ s[[Position[Reverse@ IntegerDigits[#, 2], 1][[All, 1]] ]] &, nn]]] (* _Michael De Vlieger_, Feb 25 2020 *)

%Y Cf. A000085.

%K nonn

%O 0,3

%A _Nick Krempel_, Feb 22 2020

%E More terms from _Alois P. Heinz_, Feb 27 2020