A000722 Number of invertible Boolean functions of n variables: a(n) = (2^n)!.

1, 2, 24, 40320, 20922789888000, 263130836933693530167218012160000000, 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

These are invertible maps from {0,1}^n to {0,1}^n, or in other words permutations of the 2^n binary vectors of length n.

2^n-th order derivative of n-th Mandelbrot iterate. Example: a(2) = 24, after one iterate in the Mandelbrot(z(n+1) = z(n)^2 + c) we have the function z(2) = z^4 + 2*c*z^2 + c^2 + c, for which the 4th-order derivative is 24. - Bert van den Bosch (zeusooooo(AT)hotmail.com), Sep 07 2003

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Delbert L. Johnson, Table of n, a(n) for n = 0..8
• M. A. Harrison, The number of classes of invertible Boolean functions, J. ACM 10 (1963), 25-28. [Annotated scan of page 27 only]
• C. S. Lorens, Invertible Boolean functions, IEEE Trans. Electron. Computers, EC-13 (1964), 529-541.
• C. S. Lorens, Invertible Boolean functions, IEEE Trans. Electron. Computers, EC-13 (1964), 529-541. [Annotated scan of page 530 only]
• I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
• Index entries for sequences related to Boolean functions

Crossrefs

Cf. A000652, A000653, A000654, A001038, A001537, A046856, A046857, A244061.

Programs

Mathematica

a[n_] := Factorial[2^n]; Table[a[n],{n,0,6}] (* James C. McMahon, Dec 06 2023 *)

PARI

atonfact(a,n) = {sr=0; for(x=1,n, y =(a^x)!; sr+=1.0/y; print1(y" "); ); print(); print(sr) }

Formula

a(n) = (2^n)!.

Sum of reciprocals = 0.54169146825401604874... - Cino Hilliard, Feb 08 2003