%I #20 May 20 2022 08:03:06
%S 1,1,2,4,9,18,42,91,208,470,1089,2509,5869,13730,32371,76510,181708,
%T 432635,1033656,2475384,5943395,14299532,34475030,83263872,201441431,
%U 488092897,1184353643,2877611984,7000359244,17049288304,41568056484,101449503960,247828380511
%N Number of connected functions of n points with no symmetries.
%H Alois P. Heinz, <a href="/A032175/b032175.txt">Table of n, a(n) for n = 1..2503</a> (first 500 terms from Andrew Howroyd)
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F "CHK" (necklace, identity, unlabeled) transform of A004111.
%p g:= proc(n) option remember; `if`(n<2, n, add(g(n-k)*add(g(d)*d*
%p (-1)^(k/d+1), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> g(n)+b(n, n-1):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, May 19 2022
%t g[n_] := g[n] = If[n < 2, n, Sum[g[n - k]*Sum[g[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1)];
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[j - 1 - a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
%t a[n_] := g[n] + b[n, n - 1];
%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, May 20 2022, after _Alois P. Heinz_ *)
%o (PARI) \\ here IdTreeGf is g.f. of A004111.
%o IdTreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
%o seq(n)={Vec(CHK(IdTreeGf(n), n))} \\ _Andrew Howroyd_, Aug 31 2018
%Y Cf. A002861, A004111.
%K nonn
%O 1,3
%A _Christian G. Bower_