%I #18 Feb 23 2019 10:13:16
%S 0,1,6,57,614,7284,91566,1200705,16232820,224669049,3167546164,
%T 45332076360,656839389030,9616669678368,142047629163678,
%U 2114254565121153,31678697416852592,477435309876933207,7232846873430148224,110080098299375941607,1682316692754904517268
%N Number of rooted trees with n nodes and 6-colored non-root nodes.
%H Alois P. Heinz, <a href="/A246235/b246235.txt">Table of n, a(n) for n = 0..500</a>
%H Loïc Foissy, <a href="https://arxiv.org/abs/1811.07572">Algebraic structures on typed decorated rooted trees</a>, arXiv:1811.07572 [math.RA], 2018.
%F a(n) ~ c * d^n / n^(3/2), where d = 16.50220884469300156571122110433906696332..., c = 0.06729936839826481812867613714374483... . - _Vaclav Kotesovec_, Aug 20 2014
%F G.f. A(x) satisfies: A(x) = x*exp(6*Sum_{k>=1} A(x^k)/k). - _Ilya Gutkovskiy_, Mar 20 2018
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n<2, n, (add(add(d*
%p a(d), d=divisors(j))*a(n-j)*6, j=1..n-1))/(n-1))
%p end:
%p seq(a(n), n=0..25);
%t a[n_] := a[n] = If[n<2, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j]*6, {j, 1, n-1}]/(n-1)];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 23 2019, translated from Maple *)
%Y Column k=6 of A242249.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Aug 19 2014