cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

1, 2, 6, 44, 2064, 4292864, 18430828806144, 339695459704759501186924544, 115393005344028056118476170527365821430429589033713664, 13315545682326887517994506072805639054664915214679444711916992466809542959290217586307654871548759705124864
Offset: 0

Views

Author

Nick Krempel, Feb 22 2020

Keywords

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

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...

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

Original entry on oeis.org

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

Views

Author

Nick Krempel, Feb 27 2020

Keywords

Comments

Bisection of A332840.

Examples

			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).

Links

Crossrefs

Cf. A332758, A332840.

Programs

  • 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 *)
  • 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

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).

Extensions

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

A332840 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

Original entry on oeis.org

1, 1, 3, 3, 17, 17, 51, 51, 417, 417, 1251, 1251, 7089, 7089, 21267, 21267, 206657, 206657, 619971, 619971, 3513169, 3513169, 10539507, 10539507, 86175969, 86175969, 258527907, 258527907, 1464991473, 1464991473, 4394974419, 4394974419, 44854599297, 44854599297, 134563797891
Offset: 0

Views

Author

Nick Krempel, Feb 26 2020

Keywords

Comments

As a Sylow 2-subgroup of S_(4n+2) is isomorphic to a Sylow 2-subgroup of S_(4n) direct product C_2, the terms of this sequence come in equal pairs.
Also the number of fixed-point free involutory automorphisms of the full binary tree with 2n leaves (hence 4n-1 vertices) in which all left children are complete (perfect) binary trees.

Examples

			For n=2, the a(2)=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).

Crossrefs

Cf. A123023, A001147, A332758, A332869.

Programs

  • 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), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..40);  # 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], {k, Flatten@ Position[ Reverse@ IntegerDigits[ n, 2], 1]}];
a /@ Range[0, 34] (* Jean-François Alcover, Apr 10 2020 *)
  • PARI
    a(n)={my(v=vector(logint(max(1,n), 2)+1)); v[1]=1; 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

Formula

a(n) = Product(A332758(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 = 2.1522868238..., 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).
a(n) = A332869(floor(n/2)). - Andrew Howroyd, Feb 27 2020

Extensions

Terms a(18) and beyond from Andrew Howroyd, Feb 27 2020
Showing 1-3 of 3 results.

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