A141318 Euler transform of A025225.

1, 2, 7, 28, 139, 762, 4549, 28464, 184742, 1230348, 8362298, 57756888, 404218046, 2860222020, 20428192842, 147072637472, 1066221149519, 7776835940430, 57028407361537, 420199462347876, 3109418928328473, 23098206138595710, 172184651286785655, 1287629984135461888

Offset: 0

Author

Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

Keywords

• nonn

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.

Crossrefs

Cf. A025225.

Programs

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 *)

Formula

a(n) ~ c * 8^n / n^(3/2), where c = 0.23667440081254669364903300472... . - Vaclav Kotesovec, Sep 04 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 28 2017