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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2135

Mathematics Stack Exchange Counting free trees from rooted trees.

Formula

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

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]

Crossrefs

Cf. A000055, A000081.

Sequence in context: A003006 A052321 A114584 * A039826 A221547 A289471

Adjacent sequences: A328776 A328777 A328778 * A328780 A328781 A328782

Keyword

nonn

Author

Geoffrey Critzer, Jul 06 2020

Status

approved