OFFSET
1,1
COMMENTS
Number of generators of degree n of the primitive Lie algebra of the Hopf algebra of 2-colored planar binary trees.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..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.
MAPLE
with(numtheory):
b:= proc(n) option remember;
`if`(n=0, 1, add(add((2^d)*binomial(2*d-2, d-1),
d=divisors(j)) *b(n-j), j=1..n)/n)
end:
a:= proc(n) option remember;
`if`(n<1, -1, -add(a(n-i) *b(i), i=1..n))
end:
seq(a(n), n=1..30); # Alois P. Heinz, Jan 27 2012
MATHEMATICA
b[n_] := b[n] = If[n==0, 1, Sum[Sum[2^d*Binomial[2*d-2, d-1], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; a[n_] := a[n] = If[n<1, -1, -Sum[a[n-i]* b[i], {i, 1, 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
STATUS
approved