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A332757
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Number of involutions (plus identity) in the n-fold iterated wreath product of C_2.
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3
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1, 2, 6, 44, 2064, 4292864, 18430828806144, 339695459704759501186924544, 115393005344028056118476170527365821430429589033713664, 13315545682326887517994506072805639054664915214679444711916992466809542959290217586307654871548759705124864
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 1.
a(n) ~ C^(2^n) for C = 1.611662639909645505576094683462403213269601342091954838587...
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EXAMPLE
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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).
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MATHEMATICA
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Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {1}, 9] (* Michael De Vlieger, Feb 25 2020 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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