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%I #16 Jan 04 2021 08:43:44
%S 0,2,10,44,196,876,4020,18766,89322,431758,2116220,10494080,52569504,
%T 265647586,1352621168,6933127446,35745747902,185256755454,
%U 964575991660,5043194697556,26467075595080,139375175511598,736228488297566,3900073083063348,20714052518640904
%N Number of rooted trees with n nodes colored using exactly 2 colors.
%H Andrew Howroyd, <a href="/A339642/b339642.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) = A038055(n) - 2*A000081(n).
%F a(n) = 2*(A000151(n) - A000081(n)).
%e a(3) = 10 includes 5 trees and their color complements:
%e (1(12)), (1(22)), (1(1(2))), (1(2(1))), (1(2(2))).
%p b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
%p d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
%p end:
%p a:= n-> b(n, 2)-2*b(n, 1):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Dec 11 2020
%t 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)];
%t a[n_] := b[n, 2] - 2*b[n, 1];
%t Array[a, 25] (* _Jean-François Alcover_, Jan 04 2021, after _Alois P. Heinz_ *)
%o (PARI) \\ See A141610 for U(N,m)
%o seq(n)={U(n,2) - 2*U(n,1)}
%Y Column 2 of A141610.
%Y Cf. A000081, A000151, A038055, A339643.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Dec 11 2020
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