A032175 Number of connected functions of n points with no symmetries.

1, 1, 2, 4, 9, 18, 42, 91, 208, 470, 1089, 2509, 5869, 13730, 32371, 76510, 181708, 432635, 1033656, 2475384, 5943395, 14299532, 34475030, 83263872, 201441431, 488092897, 1184353643, 2877611984, 7000359244, 17049288304, 41568056484, 101449503960, 247828380511

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2503 (first 500 terms from Andrew Howroyd)
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A002861, A004111.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, n, add(g(n-k)*add(g(d)*d*
       (-1)^(k/d+1), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> g(n)+b(n, n-1):
seq(a(n), n=1..40);  # Alois P. Heinz, May 19 2022

Mathematica

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)];
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}]]];
a[n_] := g[n] + b[n, n - 1];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

PARI

\\ here IdTreeGf is g.f. of A004111.
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)}
CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={Vec(CHK(IdTreeGf(n), n))} \\ Andrew Howroyd, Aug 31 2018

Formula

"CHK" (necklace, identity, unlabeled) transform of A004111.