OFFSET
0,2
COMMENTS
Dimensions of the graded components of the domain of cocommutativity of the Hopf algebra of 2-colored planar binary trees.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008.
FORMULA
a(n) ~ c * 8^n / n^(3/2), where c = 0.23667440081254669364903300472... . - Vaclav Kotesovec, Sep 04 2014
MAPLE
with(numtheory):
a:= proc(n) option remember;
`if`(n=0, 1, add(add((2^d)*binomial(2*d-2, d-1),
d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jan 27 2012
MATHEMATICA
a[n_] := a[n] = If[n==0, 1, Sum[Sum[2^d*Binomial[2*d-2, d-1], {d, Divisors[ j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jul 28 2017
STATUS
approved