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A332869 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 4n.

%I A332869 #14 Apr 10 2020 08:16:54
%S A332869 1,3,17,51,417,1251,7089,21267,206657,619971,3513169,10539507,
%T A332869 86175969,258527907,1464991473,4394974419,44854599297,134563797891,
%U A332869 762528188049,2287584564147,18704367906849,56113103720547,317974254416433,953922763249299,9269516926920129
%N A332869 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 4n.
%C A332869 Bisection of A332840.
%H A332869 Alois P. Heinz, <a href="/A332869/b332869.txt">Table of n, a(n) for n = 0..1502</a>
%F A332869 a(n) = A332840(2*n).
%F A332869 a(n) = Product(A332758(k+2)) where k ranges over the positions of 1 bits in the binary expansion of n.
%F A332869 a(n) = big-Theta(C^n) for C = 4.63233857..., 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).
%e A332869 For n=1, the a(1)=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 A332869 b:= proc(n) b(n):=`if`(n=0, 0, b(n-1)^2+2^(2^(n-1)-1)) end:
%p A332869 a:= n-> (l-> mul(`if`(l[i]=1, b(i+1), 1), i=1..nops(l)))(Bits[Split](n)):
%p A332869 seq(a(n), n=0..32);  # _Alois P. Heinz_, Feb 27 2020
%t A332869 A332758[n_] := A332758[n] = If[n==0, 0, A332758[n-1]^2 + 2^(2^(n-1)-1)];
%t A332869 a[n_] := Product[A332758[k+1], {k, Flatten@ Position[ Reverse@ IntegerDigits[n, 2], 1]}];
%t A332869 a /@ Range[0, 24] (* _Jean-François Alcover_, Apr 10 2020 *)
%o A332869 (PARI) a(n)={my(v=vector(logint(max(1,n), 2)+2)); v[1]=1; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=2, #v, if(bittest(n,k-2), v[k], 1))} \\ _Andrew Howroyd_, Feb 27 2020
%Y A332869 Cf. A332758, A332840.
%K A332869 nonn
%O A332869 0,2
%A A332869 _Nick Krempel_, Feb 27 2020
%E A332869 Terms a(9) and beyond from _Andrew Howroyd_, Feb 27 2020