|
%I #11 Apr 14 2019 07:51:06
%S 0,1,8,92,1304,20198,332520,5703724,100847976,1824927697,33634879304,
%T 629201396744,11915930584384,228010216559592,4401559021963488,
%U 85616787777724400,1676436841812675760,33017479163392717192,653643628799220208104,12999812350464606307796
%N Number of rooted identity trees with n nodes and 8-colored non-root nodes.
%H Alois P. Heinz, <a href="/A255520/b255520.txt">Table of n, a(n) for n = 0..750</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 21.5622387024302370660187831154056800411286761376313324441779580180359..., c = 0.049440632575743414117260362085656158155861722... . - _Vaclav Kotesovec_, Feb 24 2015
%F From _Ilya Gutkovskiy_, Apr 14 2019: (Start)
%F G.f. A(x) satisfies: A(x) = x*exp(8*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(8*a(n)). (End)
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
%p 8*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
%p end:
%p seq(a(n), n=0..30);
%Y Column k=8 of A255517.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Feb 24 2015
| 