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A255519 Number of rooted identity trees with n nodes and 7-colored non-root nodes.

%I A255519 #12 Apr 13 2019 22:19:51
%S A255519 0,1,7,70,868,11746,169022,2533895,39156244,619261762,9974917309,
%T A255519 163078496420,2699109217420,45136794135853,761492594770649,
%U A255519 12944937176754422,221518183089901621,3812823547509811611,65966582592220602621,1146568019260356348978
%N A255519 Number of rooted identity trees with n nodes and 7-colored non-root nodes.
%H A255519 Alois P. Heinz, <a href="/A255519/b255519.txt">Table of n, a(n) for n = 0..750</a>
%F A255519 a(n) ~ c * d^n / n^(3/2), where d = 18.843901825822305757579605844910623225182677164912157727486865738745..., c = 0.056432508184666421552433382560980970822794... . - _Vaclav Kotesovec_, Feb 24 2015
%F A255519 From _Ilya Gutkovskiy_, Apr 13 2019: (Start)
%F A255519 G.f. A(x) satisfies: A(x) = x*exp(7*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
%F A255519 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(7*a(n)). (End)
%p A255519 with(numtheory):
%p A255519 a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
%p A255519       7*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
%p A255519     end:
%p A255519 seq(a(n), n=0..30);
%Y A255519 Column k=7 of A255517.
%K A255519 nonn
%O A255519 0,3
%A A255519 _Alois P. Heinz_, Feb 24 2015