A373796 a(n) = Product_{k=1..n} k^Stirling_2(n,k).

1, 1, 2, 24, 373248, 145563074713240071045120, 4671362199215574200933052290575558394040074468464419088211590760845408889948035734306816000000000000000

Offset: 0

Author

N. J. A. Sloane, Jul 07 2024, based on an email from Don Knuth, Jul 06 2024

Keywords

• nonn

Comments

a(n) is the number of n-ary clones of the "discriminator function" t(x,y,z) defined by t(x,y,z)=x if x != y, t(x,x,z)=z.

For example, one of the 24 clones when n=3 is the function f(x,y,z)=t(t(y,z,x),x,t(x,y,z)), which has the property that f(x,x,x)=x, f(x,x,y)=y, f(x,y,x)=y, f(x,y,y)=x, f(x,y,z)=y when x,y,z are distinct. There are 24 meaningful ways to specify the right-hand sides of these five equations, and each of those functions can be expressed as a term in t.

There are a(4) meaningful ways to specify the right-hand sides of A000110(4)=15 analogous equations for a four-parameter function, and so on. - Don Knuth, Jul 07 2024

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..7
• Alden F. Pixley, The Ternary Discriminator Function in Universal Algebra, Mathematische Annalen, 191 (1971), 167-180.

Crossrefs

Cf. A008277, A319761.

Programs

Maple

a:= n-> mul(k^Stirling2(n,k), k=1..n):
seq(a(n), n=0..6);  # Alois P. Heinz, Jan 30 2025

Mathematica

A373796[n_] := Product[k^StirlingS2[n, k], {k, n}];
Array[A373796, 8, 0] (* Paolo Xausa, Jul 10 2024 *)

PARI

a(n)=prod(k=1,n,k^stirling(n,k,2)) \\ Hugo Pfoertner, Jul 07 2024