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Number of rooted trees with n nodes and 8-colored non-root nodes.
2

%I #17 Feb 23 2019 10:13:10

%S 0,1,8,100,1432,22570,377320,6578116,118238600,2175619923,40778137032,

%T 775828919936,14944103723856,290858342628604,5711285455910096,

%U 113005043943326568,2250850657029983808,45095294493866921469,908159403846847306568,18373705506139825769712

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

%H Alois P. Heinz, <a href="/A246237/b246237.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 = 21.9366222112987115910888213763759058905..., c = 0.05031446862451857508141944218348994381... . - _Vaclav Kotesovec_, Aug 26 2014

%F G.f. A(x) satisfies: A(x) = x*exp(8*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)*8, 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]*8, {j, 1, n-1}]/(n-1)];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 23 2019, from Maple *)

%Y Column k=8 of A242249.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 19 2014