A025225 - OEIS

%I #65 Sep 08 2022 08:44:48

%S 2,4,16,80,448,2688,16896,109824,732160,4978688,34398208,240787456,

%T 1704034304,12171673600,87636049920,635361361920,4634400522240,

%U 33985603829760,250420238745600,1853109766717440,13765958267043840,102618961627054080,767411365211013120

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = (2^n)*C(n-1), where C = A000108 (Catalan numbers).

%C Number of generators of degree n of the Hopf algebra of 2-colored planar binary trees. Also, dimensions of the graded components of the primitive Lie algebra of the same Hopf algebra. - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

%H Alois P. Heinz, <a href="/A025225/b025225.txt">Table of n, a(n) for n = 1..350</a>

%H Suzanne Bobzien, <a href="http://web.archive.org/web/20150521010834/http://philpapers.org/archive/BOBTCO-2.pdf">The combinatorics of the Stoic conjunction: Hipparchus refuted and Chrysippus vindicated</a>, Oxford Studies in Ancient Philosophy, Vol. XL, Summer 2011, pp. 157-188.

%H L. Guo and W. Y. Sit, <a href="http://dx.doi.org/10.1007/s11786-010-0061-2">Enumeration and generating functions of Rota-Baxter Words</a>, Math. Comput. Sci. 4 (2010) 313-337, remark 3.3.

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=653">Encyclopedia of Combinatorial Structures 653</a>.

%H 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.

%H Volkan Yildiz, <a href="http://arxiv.org/abs/1203.4645">Counting false entries in truth tables of bracketed formulas connected by m-implication</a>, arXiv:1203.4645 [math.CO], 2012.

%H Volkan Yildiz, <a href="http://arxiv.org/abs/1205.5595">General combinatorical structure of truth tables of bracketed formulas connected by implication</a>, arXiv:1205.5595 [math.CO], 2012.

%F G.f.: (1-sqrt(1-8*x))/2. - _Michael Somos_, Jun 08 2000

%F Given g.f. C(x) and given A(x)= g.f. of A100238, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - _Michael Somos_, Sep 07 2005

%F n*a(n) + 4*(-2*n+3)*a(n-1) = 0. - _R. J. Mathar_, Feb 25 2015

%p a:= n-> (2^n)*binomial(2*n-2, n-1)/n:

%p seq(a(n), n=1..25); # _Alois P. Heinz_, Jan 27 2012

%t InverseSeries[Series[y/2-y^2/2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* _Len Smiley_, Apr 13 2000 *)

%t a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 1, 23}] (* _Jean-François Alcover_, Jul 09 2013 *)

%o (PARI) a(n)=polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)

%o (Magma) [2^n*Catalan(n-1): n in [1..30]]; // _Vincenzo Librandi_, Nov 06 2016

%Y Essentially identical to A115125.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_

%E Typo in definition corrected by _R. J. Mathar_, Aug 11 2008