A339642 Number of rooted trees with n nodes colored using exactly 2 colors.

0, 2, 10, 44, 196, 876, 4020, 18766, 89322, 431758, 2116220, 10494080, 52569504, 265647586, 1352621168, 6933127446, 35745747902, 185256755454, 964575991660, 5043194697556, 26467075595080, 139375175511598, 736228488297566, 3900073083063348, 20714052518640904

Offset: 1

Author

Andrew Howroyd, Dec 11 2020

Keywords

• nonn

Examples

			a(3) = 10 includes 5 trees and their color complements:
   (1(12)), (1(22)), (1(1(2))), (1(2(1))), (1(2(2))).
		

Links

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

Crossrefs

Column 2 of A141610.

Cf. A000081, A000151, A038055, A339643.

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, 2)-2*b(n, 1):
seq(a(n), n=1..25);  # 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, 2] - 2*b[n, 1];
Array[a, 25] (* Jean-François Alcover, Jan 04 2021, after Alois P. Heinz *)

PARI

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

Formula

a(n) = A038055(n) - 2*A000081(n).

a(n) = 2*(A000151(n) - A000081(n)).