cp's OEIS Frontend

A332868 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

%I A332868 #12 Apr 10 2020 15:16:15
%S A332868 1,2,6,12,44,88,264,528,2064,4128,12384,24768,90816,181632,544896,
%T A332868 1089792,4292864,8585728,25757184,51514368,188886016,377772032,
%U A332868 1133316096,2266632192,8860471296,17720942592,53162827776,106325655552,389860737024,779721474048,2339164422144
%N A332868 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.
%C A332868 Bisection of A332759.
%H A332868 Alois P. Heinz, <a href="/A332868/b332868.txt">Table of n, a(n) for n = 0..2412</a>
%F A332868 a(n) = A332759(2*n).
%F A332868 a(n) = Product(A332757(k+1)) where k ranges over the positions of 1 bits in the binary expansion of n.
%F A332868 a(n) = big-Theta(C^n) for C = 2.59745646488..., 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 A332868 For n=2, the a(2)=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 A332868 b:= proc(n) b(n):=`if`(n=0, 1, b(n-1)^2+2^(2^(n-1)-1)) end:
%p A332868 a:= n-> (l-> mul(`if`(l[i]=1, b(i), 1), i=1..nops(l)))(Bits[Split](n)):
%p A332868 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 27 2020
%t A332868 b[n_] := b[n] = If[n == 0, 1, b[n - 1]^2 + 2^(2^(n - 1) - 1)];
%t A332868 a[n_] := Function[l, Product[If[l[[i]] == 1, b[i], 1], {i, 1, Length[l]}]][ Reverse @ IntegerDigits[n, 2]];
%t A332868 a /@ Range[0, 35] (* _Jean-François Alcover_, Apr 10 2020, after _Alois P. Heinz_ *)
%o A332868 (PARI) a(n)={my(v=vector(logint(max(1,n), 2)+1)); v[1]=2; 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 A332868 Cf. A332757, A332759.
%K A332868 nonn
%O A332868 0,2
%A A332868 _Nick Krempel_, Feb 27 2020
%E A332868 Terms a(17) and beyond from _Andrew Howroyd_, Feb 27 2020