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A003645
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a(n) = 2^n * C(n+1), where C(n) = A000108(n) Catalan numbers.
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21
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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, 191852841302753280, 1438896309770649600
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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2*a(n) matches the odd-indexed terms of A090375.
a(n) = 2^n * binomial(2n+3, n+1) / (2n+3). - Len Smiley, Feb 24 2006
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). (End)
D-finite with recurrence (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
Sum_{n>=0} 1/a(n) = 38/49 + 192*arcsin(sqrt(1/8))/(49*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 14/27 + 32*log(2)/81. (End)
a(n) = Product_{1 <= i <= j <= n} (i + j + 2)/(i + j - 1). Cf. A001700. - Peter Bala, Feb 22 2023
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MAPLE
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MATHEMATICA
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Table[2^n CatalanNumber[n+1], {n, 0, 20}] (* Harvey P. Dale, May 07 2013 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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