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Number of rooted identity trees with n nodes and 7-colored non-root nodes.
2

%I #12 Apr 13 2019 22:19:51

%S 0,1,7,70,868,11746,169022,2533895,39156244,619261762,9974917309,

%T 163078496420,2699109217420,45136794135853,761492594770649,

%U 12944937176754422,221518183089901621,3812823547509811611,65966582592220602621,1146568019260356348978

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

%H Alois P. Heinz, <a href="/A255519/b255519.txt">Table of n, a(n) for n = 0..750</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 18.843901825822305757579605844910623225182677164912157727486865738745..., c = 0.056432508184666421552433382560980970822794... . - _Vaclav Kotesovec_, Feb 24 2015

%F From _Ilya Gutkovskiy_, Apr 13 2019: (Start)

%F G.f. A(x) satisfies: A(x) = x*exp(7*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)^(7*a(n)). (End)

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(

%p 7*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=7 of A255517.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 24 2015