A025225 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).

2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..350
• Suzanne Bobzien, The combinatorics of the Stoic conjunction: Hipparchus refuted and Chrysippus vindicated, Oxford Studies in Ancient Philosophy, Vol. XL, Summer 2011, pp. 157-188.
• L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337, remark 3.3.
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 653.
• 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.
• Volkan Yildiz, Counting false entries in truth tables of bracketed formulas connected by m-implication, arXiv:1203.4645 [math.CO], 2012.
• Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

Crossrefs

Essentially identical to A115125.

Cf. A052701.

Programs

Magma

[2^n*Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 06 2016

Maple

a:= n-> (2^n)*binomial(2*n-2, n-1)/n:
seq(a(n), n=1..25); # Alois P. Heinz, Jan 27 2012

Mathematica

InverseSeries[Series[y/2-y^2/2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 13 2000 *)
a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Jul 09 2013 *)

PARI

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

Formula

G.f.: (1-sqrt(1-8*x))/2. - Michael Somos, Jun 08 2000

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

n*a(n) + 4*(-2*n+3)*a(n-1) = 0. - R. J. Mathar, Feb 25 2015

a(n) = 2*A052701(n). - Alois P. Heinz, Feb 16 2025

Extensions

Typo in definition corrected by R. J. Mathar, Aug 11 2008