A246237 Number of rooted trees with n nodes and 8-colored non-root nodes.

0, 1, 8, 100, 1432, 22570, 377320, 6578116, 118238600, 2175619923, 40778137032, 775828919936, 14944103723856, 290858342628604, 5711285455910096, 113005043943326568, 2250850657029983808, 45095294493866921469, 908159403846847306568, 18373705506139825769712

Offset: 0

Author

Alois P. Heinz, Aug 19 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.

Crossrefs

Column k=8 of A242249.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      a(d), d=divisors(j))*a(n-j)*8, j=1..n-1))/(n-1))
    end:
seq(a(n), n=0..25);

Mathematica

a[n_] := a[n] = If[n<2, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j]*8, {j, 1, n-1}]/(n-1)];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 23 2019, from Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 21.9366222112987115910888213763759058905..., c = 0.05031446862451857508141944218348994381... . - Vaclav Kotesovec, Aug 26 2014

G.f. A(x) satisfies: A(x) = x*exp(8*Sum_{k>=1} A(x^k)/k). - Ilya Gutkovskiy, Mar 20 2018