A003181 Number of P-equivalence classes of nondegenerate Boolean functions of n variables.

2, 2, 8, 68, 3904, 37329264, 25626412300941056, 67516342973185974302549277749387264, 2871827610052485009904013737758920847602293486924450772201235462734479360

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also the number of non-isomorphic sets of subsets of {1..n} with union {1..n}. - Gus Wiseman, Aug 05 2019

Examples

			From _Gus Wiseman_, Aug 05 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{1},{2}}
                  {{},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)
		

References

• S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..12
• S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971. [Annotated scans of a few pages]
• Index entries for sequences related to Boolean functions

Crossrefs

Cf. A000371, A001146, A003180, A003465, A055621, A007537, A326881.

Programs

Maple

b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
      h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
    end:
a:= n-> `if`(n=0, 2, b(n$2, [])-b(n-1$2, [])):
seq(a(n), n=0..8);  # Alois P. Heinz, Aug 14 2019

Mathematica

b[n_, i_, l_] := If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
a[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]];
a /@ Range[0, 8] (* Jean-François Alcover, Apr 11 2020, after Alois P. Heinz *)

Formula

a(n) = A003180(n) - A003180(n-1), for n >= 1. - Christian Sievers, Jul 22 2016

a(n) = 2 * A055621(n). - Gus Wiseman, Aug 05 2019

Extensions

More terms from Christian Sievers, Jul 22 2016

Definition clarified by Ivo Timoteo, Mar 14 2017