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A246237 Number of rooted trees with n nodes and 8-colored non-root nodes.

%I A246237 #17 Feb 23 2019 10:13:10
%S A246237 0,1,8,100,1432,22570,377320,6578116,118238600,2175619923,40778137032,
%T A246237 775828919936,14944103723856,290858342628604,5711285455910096,
%U A246237 113005043943326568,2250850657029983808,45095294493866921469,908159403846847306568,18373705506139825769712
%N A246237 Number of rooted trees with n nodes and 8-colored non-root nodes.
%H A246237 Alois P. Heinz, <a href="/A246237/b246237.txt">Table of n, a(n) for n = 0..500</a>
%H A246237 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 A246237 a(n) ~ c * d^n / n^(3/2), where d = 21.9366222112987115910888213763759058905..., c = 0.05031446862451857508141944218348994381... . - _Vaclav Kotesovec_, Aug 26 2014
%F A246237 G.f. A(x) satisfies: A(x) = x*exp(8*Sum_{k>=1} A(x^k)/k). - _Ilya Gutkovskiy_, Mar 20 2018
%p A246237 with(numtheory):
%p A246237 a:= proc(n) option remember; `if`(n<2, n, (add(add(d*
%p A246237       a(d), d=divisors(j))*a(n-j)*8, j=1..n-1))/(n-1))
%p A246237     end:
%p A246237 seq(a(n), n=0..25);
%t A246237 a[n_] := a[n] = If[n<2, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j]*8, {j, 1, n-1}]/(n-1)];
%t A246237 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 23 2019, from Maple *)
%Y A246237 Column k=8 of A242249.
%K A246237 nonn
%O A246237 0,3
%A A246237 _Alois P. Heinz_, Aug 19 2014