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A003645
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2^n*C(n+1), where C(n) = A000108(n) Catalan numbers.
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13
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1, 4, 20, 112, 672, 4224, 27456, 183040, 1244672, 8599552, 60196864, 426008576, 3042918400, 21909012480, 158840340480, 1158600130560, 8496400957440, 62605059686400, 463277441679360, 3441489566760960, 25654740406763520
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (liskov(AT)im.bas-net.by), 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 (rhhardin(AT)att.net), 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. - Wen-jin Woan (wwoan(AT)howard.edu), Oct 24 2004
The number of rooted bipartite n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), 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. [From David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010]
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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. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 652
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.
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FORMULA
| a(n)=(2^n)*binomial(2*n+3, n+1)/(2*n+3) - Len Smiley (smiley(AT)math.uaa.alaska.edu)
G.f.: (1-4x-sqrt(1-8x))/(8x^2) = C(2x)^2 where C(x) is 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
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PROG
| (PARI) a(n)=if(n<0, 0, 2^n*(2*n+2)!/(n+1)!/(n+2)!)
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CROSSREFS
| Cf. A069724, A069725. a(n)=A052701(n+2)/2.
Third row of array A102539.
Column of array A073165.
Cf. A052701.
Sequence in context: A081335 A136783 A080609 * A081085 A192624 A108447
Adjacent sequences: A003642 A003643 A003644 * A003646 A003647 A003648
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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