A255521 - OEIS

%I #11 Apr 14 2019 07:51:12

%S 0,1,9,117,1866,32553,603414,11654634,232034283,4728048201,

%T 98125181461,2066983603704,44079196497075,949772378078829,

%U 20645820782745363,452215682045713701,9970925646977589555,221133330528834114000,4929622717525248345174,110400838255998014848137

%N Number of rooted identity trees with n nodes and 9-colored non-root nodes.

%H Alois P. Heinz, <a href="/A255521/b255521.txt">Table of n, a(n) for n = 0..700</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 24.2805556948066926165789325334976292249076194687965619357813839307368..., c = 0.04399000859622510673129847184312171422452194... . - _Vaclav Kotesovec_, Feb 24 2015

%F From _Ilya Gutkovskiy_, Apr 14 2019: (Start)

%F G.f. A(x) satisfies: A(x) = x*exp(9*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)^(9*a(n)). (End)

%p with(numtheory):

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

%p       9*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=9 of A255517.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 24 2015