cp's OEIS Frontend

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

%I A339643 #15 Jan 04 2021 08:43:30
%S A339643 0,0,9,102,870,6744,50421,371676,2731569,20113005,148752507,
%T A339643 1106207331,8274878880,62263100994,471138360426,3584051515209,
%U A339643 27399942354822,210432444531798,1622954350900455,12565580096217270,97634810663895132,761110656740387865,5951117699678438271
%N A339643 Number of rooted trees with n nodes colored using exactly 3 colors.
%H A339643 Andrew Howroyd, <a href="/A339643/b339643.txt">Table of n, a(n) for n = 1..500</a>
%F A339643 a(n) = A038059(n) - 3*A038055(n) + 3*A000081(n).
%F A339643 a(n) = 3*(A006964(n) - 2*A000151(n) + A000081(n)).
%p A339643 b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
%p A339643       d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
%p A339643     end:
%p A339643 a:= n-> b(n, 3)-3*b(n, 2)+3*b(n, 1):
%p A339643 seq(a(n), n=1..23);  # _Alois P. Heinz_, Dec 11 2020
%t A339643 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 A339643 a[n_] := b[n, 3] - 3 b[n, 2] + 3 b[n, 1];
%t A339643 Array[a, 23] (* _Jean-François Alcover_, Jan 04 2021, after _Alois P. Heinz_ *)
%o A339643 (PARI) \\ See A141610 for U(N,m)
%o A339643 seq(n)={U(n,3) - 3*U(n,2) + 3*U(n,1)}
%Y A339643 Column 3 of A141610.
%Y A339643 Cf. A000081, A000151, A006964, A038055, A038059, A339642.
%K A339643 nonn
%O A339643 1,3
%A A339643 _Andrew Howroyd_, Dec 11 2020