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A255522 Number of rooted identity trees with n nodes and 10-colored non-root nodes.

%I A255522 #11 Apr 14 2019 07:51:18
%S A255522 0,1,10,145,2570,49860,1027602,22068705,488541820,11068982545,
%T A255522 255437694060,5983042467096,141873247900650,3399140192819340,
%U A255522 82160878859739650,2001070766525744725,49061025740711233080,1209873601374566796515,29990547373994063764080
%N A255522 Number of rooted identity trees with n nodes and 10-colored non-root nodes.
%H A255522 Alois P. Heinz, <a href="/A255522/b255522.txt">Table of n, a(n) for n = 0..700</a>
%F A255522 a(n) ~ c * d^n / n^(3/2), where d = 26.998860838916733933849490675388336975888308433826638445291076817..., c = 0.0396216952587990270999387393053224285... . - _Vaclav Kotesovec_, Feb 24 2015
%F A255522 From _Ilya Gutkovskiy_, Apr 14 2019: (Start)
%F A255522 G.f. A(x) satisfies: A(x) = x*exp(10*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
%F A255522 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(10*a(n)). (End)
%p A255522 with(numtheory):
%p A255522 a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
%p A255522       10*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
%p A255522     end:
%p A255522 seq(a(n), n=0..30);
%Y A255522 Column k=10 of A255517.
%K A255522 nonn
%O A255522 0,3
%A A255522 _Alois P. Heinz_, Feb 24 2015