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

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..2503 (first 500 terms from Andrew Howroyd)

C. G. Bower, Transforms (2)

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A002861, A004111.

Sequence in context: A026765 A264649 A259803 * A000678 A362037 A368422

Adjacent sequences: A032172 A032173 A032174 * A032176 A032177 A032178

Keyword

nonn

Author

Christian G. Bower

Status

approved