A332757 Number of involutions (plus identity) in the n-fold iterated wreath product of C_2.

1, 2, 6, 44, 2064, 4292864, 18430828806144, 339695459704759501186924544, 115393005344028056118476170527365821430429589033713664, 13315545682326887517994506072805639054664915214679444711916992466809542959290217586307654871548759705124864

Offset: 0

Author

Nick Krempel, Feb 22 2020

Keywords

• nonn

Comments

Also the number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2^n.

Also the number of involutory automorphisms (including identity) of the complete binary tree of height n.

Examples

			For n=2, the a(2)=6 elements satisfying x^2=1 in C_2 wr C_2 (which is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).
		

Links

• Claudia Scheimbauer and Thomas Stempfhuber, Relative field theories via relative dualizability, arXiv:2312.05051 [math.CT], 2023. See Theorem 6.7 at pages 5, 32, and 34.

Crossrefs

Cf. A332758.

Programs

Mathematica

Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {1}, 9] (* Michael De Vlieger, Feb 25 2020 *)

Formula

a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 1.

a(n) ~ C^(2^n) for C = 1.611662639909645505576094683462403213269601342091954838587...