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Number of rooted trees with n nodes and 9-colored non-root nodes.
2

%I #17 Feb 23 2019 10:13:04

%S 0,1,9,126,2028,35919,674964,13225632,267188229,5525568297,

%T 116400215521,2488984609644,53883301460607,1178671076752803,

%U 26011822261355487,578441979200964897,12948907700868521142,291569232606979152321,6599274332502483541080

%N Number of rooted trees with n nodes and 9-colored non-root nodes.

%H Alois P. Heinz, <a href="/A246238/b246238.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 = 24.654188324989308481297681284616076174..., c = 0.044677658292795889027452779147731609565... . - _Vaclav Kotesovec_, Aug 26 2014

%F G.f. A(x) satisfies: A(x) = x*exp(9*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)*9, 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]*9, {j, 1, n-1}]/(n-1)];

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

%Y Column k=9 of A242249.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 19 2014