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

1, 3, 17, 51, 417, 1251, 7089, 21267, 206657, 619971, 3513169, 10539507, 86175969, 258527907, 1464991473, 4394974419, 44854599297, 134563797891, 762528188049, 2287584564147, 18704367906849, 56113103720547, 317974254416433, 953922763249299, 9269516926920129

Offset

0,2

Comments

Bisection of A332840.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1502

Formula

a(n) = A332840(2*n).

a(n) = Product(A332758(k+2)) where k ranges over the positions of 1 bits in the binary expansion of n.

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).

Example

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).

Maple

b:= proc(n) b(n):=`if`(n=0, 0, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i+1), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 27 2020

Mathematica

A332758[n_] := A332758[n] = If[n==0, 0, A332758[n-1]^2 + 2^(2^(n-1)-1)];
a[n_] := Product[A332758[k+1], {k, Flatten@ Position[ Reverse@ IntegerDigits[n, 2], 1]}];
a /@ Range[0, 24] (* Jean-François Alcover, Apr 10 2020 *)

Prog

(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

Crossrefs

Cf. A332758, A332840.

Sequence in context: A132124 A011917 A018691 * A225727 A163943 A093418

Adjacent sequences: A332866 A332867 A332868 * A332870 A332871 A332872

Keyword

nonn

Author

Nick Krempel, Feb 27 2020

Extensions

Terms a(9) and beyond from Andrew Howroyd, Feb 27 2020

Status

approved