A069720 a(n) = 2^(n-1)*binomial(2*n-1, n).
1, 6, 40, 280, 2016, 14784, 109824, 823680, 6223360, 47297536, 361181184, 2769055744, 21300428800, 164317593600, 1270722723840, 9848101109760, 76467608616960, 594748067020800, 4632774416793600, 36135640450990080, 282202144474398720, 2206307674981662720, 17266755717247795200
Offset: 1
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
- Harlan J. Brothers, Pascal's Prism: Supplementary Material.
- Valery A. Liskovets and Timothy 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.
Crossrefs
Programs
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Haskell
a069720 n = (a000079 $ n - 1) * (a001700 $ n - 1) -- Reinhard Zumkeller, Jan 15 2015
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Magma
[2^(n-2)*Binomial(2*n, n): n in [1..25]]; // Vincenzo Librandi, Apr 14 2018
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Maple
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
Table[2^(n-1) Binomial[2n-1,n],{n,20}] (* Harvey P. Dale, Jan 20 2013 *) -
PARI
a(n) = binomial(2*n-1,n)<<(n-1) \\ Charles R Greathouse IV, Feb 06 2017
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SageMath
def A069720(n): return 2^(n-2)*binomial(2*n, n) print([A069720(n) for n in range(1,31)]) # G. C. Greubel, Jan 18 2025
Formula
a(n) = 2^(n-2)*binomial(2*n, n).
G.f.: (1-sqrt(1-8*x))/(4*x*sqrt(1-8*x)) = 2/(sqrt(1-8*x)*(1-sqrt(1-8*x))) - 1/(2*x). - Paul Barry, Sep 06 2004
D-finite with recurrence: n*a(n) - 4*(2*n-1)*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
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = 2*A082143(n-1). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 4/7 + 32*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + 16*log(2)/27. (End)
a(n) = ((2*n)!/4) * [x^n] (BesselI(0, 2*sqrt(2*x)) - 1). - G. C. Greubel, Jan 18 2025
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