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A069720
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a(n) = 2^(n-1)*binomial(2n-1, n).
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22
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1, 6, 40, 280, 2016, 14784, 109824, 823680, 6223360, 47297536, 361181184, 2769055744, 21300428800, 164317593600, 1270722723840, 9848101109760, 76467608616960, 594748067020800, 4632774416793600, 36135640450990080, 282202144474398720, 2206307674981662720, 17266755717247795200
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OFFSET
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1,2
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COMMENTS
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Number of rooted unicursal planar maps with n edges (unicursal means that exactly two nodes are of odd valency; there is an Eulerian path).
a(n) = A000079(n-1) * A001700(n-1); for n > 1: a(n) = 2*A082143(n-1). - Reinhard Zumkeller, Jan 15 2015
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
H. J. Brothers, Pascal's Prism: Supplementary Material.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.
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FORMULA
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a(n) = 2^(n-2)*binomial(2n, n).
G.f.: (1-sqrt(1-8x))/(4x*sqrt(1-8x)) = 2/(sqrt(1-8x)(1-sqrt(1-8x)))-1/(2x). - Paul Barry, Sep 06 2004
D-finite with recurrence n*a(n) + 4*(1-2n)*a(n-1) = 0. - R. J. Mathar, Apr 01 2012
E.g.f.: a(n) = n! * [x^n] (exp(4*x)*BesselI(0, 4*x) - 1)/4. - Peter Luschny, Aug 25 2012
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MAPLE
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Z:=(1-sqrt(1-2*z))*4^(n-1)/sqrt(1-2*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..20); # Zerinvary Lajos, Jan 01 2007
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MATHEMATICA
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Table[2^(n-1) Binomial[2n-1, n], {n, 20}] (* Harvey P. Dale, Jan 20 2013 *)
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PROG
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(Haskell)
a069720 n = (a000079 $ n - 1) * (a001700 $ n - 1)
-- Reinhard Zumkeller, Jan 15 2015
(PARI) a(n) = binomial(2*n-1, n)<<(n-1) \\ Charles R Greathouse IV, Feb 06 2017
(MAGMA) [2^(n-2)*Binomial(2*n, n): n in [1..25]]; // Vincenzo Librandi, Apr 14 2018
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CROSSREFS
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First superdiagonal of number array A082137.
Cf. A069724, A003584, A069723, A082143, A000079, A001700.
Sequence in context: A122471 A178397 A090041 * A005037 A081337 A316912
Adjacent sequences: A069717 A069718 A069719 * A069721 A069722 A069723
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Valery A. Liskovets, Apr 07 2002
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STATUS
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approved
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