A052701 a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.

0, 1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

A151374 shifted one place right. - Joerg Arndt, Mar 17 2011

The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

This is also the number of strings of length 2n-2 of two different types of balanced parentheses. For example, a(2) = 2, since the two possible strings of length 2 are [] and (), a(3) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][]. - Jeffrey Shallit, Jun 03 2006

Row sums of number triangle A110506. - Paul Barry, Jul 24 2005

Also row sums of triangle in A085880. - Philippe Deléham, Aug 01 2005

Row sums of number triangle A114608. - Philippe Deléham, Oct 15 2008

References

• V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
• M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
• Marek Bożejko, Maciej Dołęga, Wiktor Ejsmont, and Światosław R. Gal, Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, arXiv:2104.14530 [math.RT], 2021.
• F. Chapoton and S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv:1310.4521 [math.CO], 2013.
• Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics, Vol. 24, No. 2 (2017), Article P2.3.
• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 651.
• V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., Vol. 36, No.4 (2006), pp. 364-387.
• Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.
• Index to sequences related to reversion of series.

Crossrefs

Limit of array A102544.

Cf. A000108, A003645, A025225, A052714, A195699.

Programs

Maple

spec := [S,{B=Union(C,Z),S=Union(B,C),C=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

InverseSeries[Series[y-2*y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *)
Join[{0},Table[2^n CatalanNumber[n],{n,0,30}]] (* Harvey P. Dale, Aug 29 2015 *)

PARI

a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!/n!)

PARI

a(n)=if(n<1,0,polcoeff(serreverse(x-2*x^2+x*O(x^n)),n))

PARI

a(n)=if(n<1,0,polcoeff(2*x/(1+sqrt(1-8*x+O(x^n))),n))

Formula

a(n) = A052714(n)/n!.

a(n) = A003645(n-2)*2, n>1.

a(n) = 8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.

D-finite with recurrence: a(1)=1, (-4+8*n)*a(n) - (n+1)*a(n+1) = 0.

G.f.: (1-sqrt(1-8*x))/4 = x*C(2*x) where C(x) is g.f. for Catalan numbers, A000108.

G.f. A(x) satisfies 2*A(x)^2-A(x)+x=0, A(0)=0 and A(x)=x+2*A(x)^2=x/(1-2*A(x)). Series reversion of x-2*x^2. - Michael Somos, Sep 06 2003

a(0)=0, a(1)=1; a(n) = 2*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004

With a different offset, a(0)=1, a(n) = Sum_{k=0..n} Sum_{j=0..n} (j*C(2n-j-1, n-j)*C(j, k)*2^(n-j)/n), n>0. - Paul Barry, Jul 24 2005

The Hankel transform of a(n+1) = [1,2,8,40,224,1344,...] is 4^C(n+1,2). - Philippe Deléham, Nov 06 2007

G.f.: x + 4*x^2/(G(0)-4*x) where G(k) = k*(8*x+1) + 4*x + 2 - 2*x*(2*k+3)*(2*k+4)/G(k+1) ; (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013

a(n) ~ 8^(n-1)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Dec 04 2016

From Amiram Eldar, Mar 06 2022: (Start)

Sum_{n>=1} 1/a(n) = 68/49 + 96*arcsin(sqrt(1/8))/(49*sqrt(7)).

Sum_{n>=1} (-1)^(n+1)/a(n) = 20/27 - 16*log(2)/81. (End)

a(n) = A025225(n)/2 for n>=1. - Alois P. Heinz, Feb 16 2025

Extensions

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001

Additional comments from Michael Somos, Feb 24 2002