%I #26 Dec 19 2021 12:07:33
%S 1,1,7,74,1474,41876,1540696,68343112,3540691525,209612916303,
%T 13957423192794,1032436318269648,83993175608894096,
%U 7453446303042245261,716451740543945788671,74159075140708644544128,8223831291824019614386868,972718473204236819072891710
%N Number of nonisomorphic binary n-state automata without output under input permutations.
%C Also isomorphism classes of unordered pairs of endofunctions i.e. an unorder pair {f,g} of functions from {1,...,n} to itself. - _Christian G. Bower_, Dec 18 2003
%D F. Harary and E. Palmer, Graphical Enumeration, 1973.
%H Alois P. Heinz, <a href="/A054745/b054745.txt">Table of n, a(n) for n = 0..45</a>
%H M. A. Harrison, <a href="http://dx.doi.org/10.4153/CJM-1965-010-9">A census of finite automata</a>, Canad. J. Math., 17, No. 1, 1965, p. 110.
%F a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2=2} (fixA[s_1, s_2, ...;t_1, t_2]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!)) where fixA[...] = prod {i, j>=1} ( (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*t_j)). - _Christian G. Bower_, Dec 18 2003
%p with(numtheory):
%p b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
%p {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
%p end:
%p a:= proc(n) option remember; add(add(mul(mul(add(coeff(s, x, d)
%p *d, d=divisors(ilcm(i, j)))^(igcd(i, j)*coeff(s, x, i)*
%p coeff(t, x, j)), j=1..degree(t)), i=1..degree(s))
%p /mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t))
%p /mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))
%p , t=b(2$2)), s=b(n$2))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 15 2014
%t b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Table[Map[Function[{p}, p + j*x^i], b[n - i*j, i-1]], {j, 0, n/i}] // Flatten // Union]];
%t a[n_] := a[n] = Sum[Sum[Product[Product[With[{g = GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j]}, If[g==0, 1, Sum[Coefficient[s, x, d]*d, {d, Divisors[LCM[i, j]]}]^g]], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/
%t Product[i^Coefficient[t, x, i]Coefficient[t, x, i]!, {i, Exponent[t, x]}]/
%t Product[i^Coefficient[s, x, i]Coefficient[s, x, i]!, {i, Exponent[s, x]}], {t, b[2, 2]}], {s, b[n, n]}];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 16 2015, after _Alois P. Heinz_, updated Jan 01 2021 *)
%Y Cf. A001372, A054050, A054732, A054746.
%K nonn
%O 0,3
%A _Vladeta Jovovic_, Apr 22 2000
%E More terms from _Alois P. Heinz_, Aug 15 2014