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A141318 Euler transform of A025225.

%I A141318 #20 Jul 28 2017 09:48:12
%S A141318 1,2,7,28,139,762,4549,28464,184742,1230348,8362298,57756888,
%T A141318 404218046,2860222020,20428192842,147072637472,1066221149519,
%U A141318 7776835940430,57028407361537,420199462347876,3109418928328473,23098206138595710,172184651286785655,1287629984135461888
%N A141318 Euler transform of A025225.
%C A141318 Dimensions of the graded components of the domain of cocommutativity of the Hopf algebra of 2-colored planar binary trees.
%H A141318 Alois P. Heinz, <a href="/A141318/b141318.txt">Table of n, a(n) for n = 0..1000</a>
%H A141318 J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions</a>, arXiv:0806.3682 [math.CO], 2008.
%F A141318 a(n) ~ c * 8^n / n^(3/2), where c = 0.23667440081254669364903300472... . - _Vaclav Kotesovec_, Sep 04 2014
%p A141318 with(numtheory):
%p A141318 a:= proc(n) option remember;
%p A141318       `if`(n=0, 1, add(add((2^d)*binomial(2*d-2,d-1),
%p A141318                    d=divisors(j))*a(n-j), j=1..n)/n)
%p A141318     end:
%p A141318 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 27 2012
%t A141318 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_ *)
%Y A141318 Cf. A025225.
%K A141318 nonn
%O A141318 0,2
%A A141318 Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
%E A141318 a(0)=1 prepended by _Alois P. Heinz_, Jul 28 2017