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A332868 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n. 1
1, 2, 6, 12, 44, 88, 264, 528, 2064, 4128, 12384, 24768, 90816, 181632, 544896, 1089792, 4292864, 8585728, 25757184, 51514368, 188886016, 377772032, 1133316096, 2266632192, 8860471296, 17720942592, 53162827776, 106325655552, 389860737024, 779721474048, 2339164422144 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Bisection of A332759.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2412

FORMULA

a(n) = A332759(2*n).

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

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).

EXAMPLE

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).

MAPLE

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

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

seq(a(n), n=0..35);  # Alois P. Heinz, Feb 27 2020

MATHEMATICA

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

a[n_] := Function[l, Product[If[l[[i]] == 1, b[i], 1], {i, 1, Length[l]}]][ Reverse @ IntegerDigits[n, 2]];

a /@ Range[0, 35] (* Jean-Fran├žois Alcover, Apr 10 2020, after Alois P. Heinz *)

PROG

(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

CROSSREFS

Cf. A332757, A332759.

Sequence in context: A266005 A056744 A164859 * A261467 A180070 A177834

Adjacent sequences:  A332865 A332866 A332867 * A332869 A332870 A332871

KEYWORD

nonn

AUTHOR

Nick Krempel, Feb 27 2020

EXTENSIONS

Terms a(17) and beyond from Andrew Howroyd, Feb 27 2020

STATUS

approved

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Last modified January 17 10:31 EST 2021. Contains 340214 sequences. (Running on oeis4.)