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

%I A373796 #37 Jul 09 2025 05:04:31
%S A373796 1,1,2,24,373248,145563074713240071045120,
%T A373796 4671362199215574200933052290575558394040074468464419088211590760845408889948035734306816000000000000000
%N A373796 a(n) = Product_{k=1..n} k^Stirling_2(n,k).
%C A373796 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.
%C A373796 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.
%C A373796 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
%H A373796 Hugo Pfoertner, <a href="/A373796/b373796.txt">Table of n, a(n) for n = 0..7</a>
%H A373796 Alden F. Pixley, <a href="https://eudml.org/doc/162131">The Ternary Discriminator Function in Universal Algebra</a>, Mathematische Annalen, 191 (1971), 167-180.
%p A373796 a:= n-> mul(k^Stirling2(n,k), k=1..n):
%p A373796 seq(a(n), n=0..6);  # _Alois P. Heinz_, Jan 30 2025
%t A373796 A373796[n_] := Product[k^StirlingS2[n, k], {k, n}];
%t A373796 Array[A373796, 8, 0] (* _Paolo Xausa_, Jul 10 2024 *)
%o A373796 (PARI) a(n)=prod(k=1,n,k^stirling(n,k,2)) \\ _Hugo Pfoertner_, Jul 07 2024
%Y A373796 Cf. A008277, A319761.
%K A373796 nonn
%O A373796 0,3
%A A373796 _N. J. A. Sloane_, Jul 07 2024, based on an email from _Don Knuth_, Jul 06 2024