A332840 - OEIS

%I #19 Apr 10 2020 08:16:47

%S 1,1,3,3,17,17,51,51,417,417,1251,1251,7089,7089,21267,21267,206657,

%T 206657,619971,619971,3513169,3513169,10539507,10539507,86175969,

%U 86175969,258527907,258527907,1464991473,1464991473,4394974419,4394974419,44854599297,44854599297,134563797891

%N Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

%C As a Sylow 2-subgroup of S_(4n+2) is isomorphic to a Sylow 2-subgroup of S_(4n) direct product C_2, the terms of this sequence come in equal pairs.

%C Also the number of fixed-point free involutory automorphisms of the full binary tree with 2n leaves (hence 4n-1 vertices) in which all left children are complete (perfect) binary trees.

%F a(n) = Product(A332758(k+1)) where k ranges over the positions of 1 bits in the binary expansion of n.

%F a(n) = big-Theta(C^n) for C = 2.1522868238..., 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 a(n) = A332869(floor(n/2)). - _Andrew Howroyd_, Feb 27 2020

%e For n=2, the a(2)=3 fixed-point free involutions in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).

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

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

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 27 2020

%t A332758[n_] := A332758[n] = If[n == 0, 0, A332758[n-1]^2 + 2^(2^(n-1)-1)];

%t a[n_] := Product[A332758[k], {k, Flatten@ Position[ Reverse@ IntegerDigits[ n, 2], 1]}];

%t a /@ Range[0, 34] (* _Jean-François Alcover_, Apr 10 2020 *)

%o (PARI) a(n)={my(v=vector(logint(max(1,n), 2)+1)); v[1]=1; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=1, #v, if(bittest(n,k-1), v[k], 1))} \\ _Andrew Howroyd_, Feb 27 2020

%Y Cf. A123023, A001147, A332758, A332869.

%K nonn

%O 0,3

%A _Nick Krempel_, Feb 26 2020

%E Terms a(18) and beyond from _Andrew Howroyd_, Feb 27 2020