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Number of rooted trees with n nodes and 10-colored non-root nodes.
2

%I #17 Feb 23 2019 10:12:59

%S 0,1,10,155,2770,54465,1136402,24723000,554540590,12732651160,

%T 297795974970,7069820334023,169926110309380,4126836768095315,

%U 101114499262401970,2496432769621336865,62045482307629427354,1551083997228106913910,38976793037598171500920

%N Number of rooted trees with n nodes and 10-colored non-root nodes.

%H Alois P. Heinz, <a href="/A246239/b246239.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 = 27.3718979186642404090999595957978919036..., c = 0.04017690459295003799582996890456677644... . - _Vaclav Kotesovec_, Aug 26 2014

%F G.f. A(x) satisfies: A(x) = x*exp(10*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)*10, 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]*10, {j, 1, n-1}]/(n-1)];

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

%Y Column k=10 of A242249.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 19 2014