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

1,1

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.

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

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 *)

Prog

(PARI) a(n)=polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2, n)
(Magma) [2^n*Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 06 2016

Crossrefs

Essentially identical to A115125.

Sequence in context: A247007 A103619 A027436 * A115125 A326859 A213010

Adjacent sequences: A025222 A025223 A025224 * A025226 A025227 A025228

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

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

Status

approved