A328779 a(n) is the number of unlabeled unrooted trees (as in A000055) on n nodes with one designated node (exclusive) or one designated edge.

0, 1, 2, 3, 7, 15, 36, 85, 211, 525, 1341, 3449, 9001, 23671, 62835, 167881, 451557, 1221065, 3318737, 9059397, 24830110, 68299159, 188488448, 521737636, 1448154837, 4029712400, 11239492056, 31416403198, 87990722479, 246903542031, 694022911203, 1954012196966

Offset: 0

Author

Geoffrey Critzer, Jul 06 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2135
• Mathematics Stack Exchange Counting free trees from rooted trees.

Crossrefs

Cf. A000055, A000081.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, n, (add(b(n-j)*add(
      d*b(d), d=numtheory[divisors](j)), j=1..n-1))/(n-1))
    end:
a:= n-> b(n)+add(b(j)*b(n-j), j=0..n)/2+`if`(n::even, b(n/2)/2, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 17 2024

Mathematica

nn = 25; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol = SolveAlways[
  0 == Series[f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
r[x_] := Sum[a[n] x^n, {n, 0, nn}] /. sol; CoefficientList[Series[r[x] + r[x]^2/2 + r[x^2]/2, {x, 0, nn}], x]

Formula

O.g.f.: R(x) + R(x)^2/2 + R(x^2)/2 where R(x) is the o.g.f. for A000081.