login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003645 2^n * C(n+1), where C(n) = A000108(n) Catalan numbers. 18
1, 4, 20, 112, 672, 4224, 27456, 183040, 1244672, 8599552, 60196864, 426008576, 3042918400, 21909012480, 158840340480, 1158600130560, 8496400957440, 62605059686400, 463277441679360, 3441489566760960, 25654740406763520, 191852841302753280, 1438896309770649600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of nonisomorphic unrooted unicursal planar maps with n+2 edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency). - Valery A. Liskovets, Apr 07 2002

Total number of vertices in rooted Eulerian planar maps with n+1 edges.

Half the number of ways to dog-ear every page of an n+1 page book. - R. H. Hardin, Jun 21 2002

Convolution of A052701(n+1) with itself.

Number of Motzkin lattice paths with weights: 1 for up step, 4 for level step and 4 for down step. - Wenjin Woan, Oct 24 2004

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

Also the number of paths of length 2n+1 in a binary tree between two vertices that are one step away from each other. [David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010]

2*a(n) for n>1 is the number of increasing strict binary trees with 2n-1 nodes that simultaneously avoid 213 and 231 in the classical sense. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 22 2014

REFERENCES

L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.

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

Park, Youngja, and SeungKyung Park. "Enumeration of generalized lattice paths by string types, peaks, and ascents." Discrete Mathematics 339.11 (2016): 2652-2659.

LINKS

Table of n, a(n) for n=0..22.

A. Claesson, S. Kitaev and A. de Mier, An involution on bicubic maps and beta(0,1)-trees, arXiv preprint arXiv:1210.3219 [math.CO], 2012. - From N. J. A. Sloane, Jan 01 2013

S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 652

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

FORMULA

a(n) = 2^n * binomial(2n+3, n+1) / (2n+3). - Len Smiley

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

Let M = the following production matrix:

2, 2, 0, 0, 0,...

2, 2, 2, 0, 0,...

2, 2, 2, 2, 0,...

2, 2, 2, 2, 2,...

...

a(n) = sum of top row terms in M^n. Example: top row of M^3 = (40, 40, 24, 8, 0, 0, 0,...), sum = 112 = a(3). - Gary W. Adamson, Jul 12 2011

Conjecture: (n+2)*a(n)-4*(2n+1)*a(n-1)=0. - R. J. Mathar, Apr 01 2012

E.g.f.: a(n) = n!* [x^n] exp(4*x)*BesselI(1, 4*x)/(2*x). - Peter Luschny, Aug 25 2012

Expansion of square of continued fraction 1/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - ...)))). - Ilya Gutkovskiy, Apr 19 2017

MAPLE

A003645:=n->2^n*binomial(2*n+3, n+1)/(2*n+3): seq(A003645(n), n=0..30); # Wesley Ivan Hurt, Aug 23 2014

MATHEMATICA

Table[2^n CatalanNumber[n+1], {n, 0, 20}] (* Harvey P. Dale, May 07 2013 *)

PROG

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

(MAGMA) [2^n*Binomial(2*n+3, n+1)/(2*n+3) : n in [0..30]]; // Wesley Ivan Hurt, Aug 23 2014

CROSSREFS

Cf. A069724, A069725. a(n) = A052701(n+2)/2.

Third row of array A102539.

Column of array A073165.

Cf. A000108, A052701.

Cf. 2*a(n) is the odd indexed terms of A025235.

Sequence in context: A136783 A227726 A080609 * A081085 A212326 A192624

Adjacent sequences:  A003642 A003643 A003644 * A003646 A003647 A003648

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 17 11:40 EDT 2017. Contains 290635 sequences.