

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
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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 dogear 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 nedge 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 2n1 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), 149160.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 200501, Montreal, Canada, 2005.
Park, Youngja, and SeungKyung Park. "Enumeration of generalized lattice paths by string types, peaks, and ascents." Discrete Mathematics 339.11 (2016): 26522659.


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 OnLine 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), 209221.
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364387.
M. Z. Spivey and L. L. Steil, The kBinomial 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.: (14xsqrt(18x))/(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(n1)=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



