A339643 Number of rooted trees with n nodes colored using exactly 3 colors.

0, 0, 9, 102, 870, 6744, 50421, 371676, 2731569, 20113005, 148752507, 1106207331, 8274878880, 62263100994, 471138360426, 3584051515209, 27399942354822, 210432444531798, 1622954350900455, 12565580096217270, 97634810663895132, 761110656740387865, 5951117699678438271

Offset: 1

Author

Andrew Howroyd, Dec 11 2020

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Column 3 of A141610.

Cf. A000081, A000151, A006964, A038055, A038059, A339642.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
      d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
    end:
a:= n-> b(n, 3)-3*b(n, 2)+3*b(n, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Dec 11 2020

Mathematica

b[n_, k_] := b[n, k] = If[n < 2, k*n, (Sum[Sum[b[d, k]*d, {d, Divisors[j]}]*b[n - j, k], {j, 1, n - 1}])/(n - 1)];
a[n_] := b[n, 3] - 3 b[n, 2] + 3 b[n, 1];
Array[a, 23] (* Jean-François Alcover, Jan 04 2021, after Alois P. Heinz *)

PARI

\\ See A141610 for U(N,m)
seq(n)={U(n,3) - 3*U(n,2) + 3*U(n,1)}

Formula

a(n) = A038059(n) - 3*A038055(n) + 3*A000081(n).

a(n) = 3*(A006964(n) - 2*A000151(n) + A000081(n)).