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A063016
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a(n) is the product of Catalan(n) and (2^(n+1) - 1).
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2
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1, 3, 14, 75, 434, 2646, 16764, 109395, 730730, 4973826, 34381412, 240728670, 1703826292, 12170930700, 87633375480, 635351667075, 4634365164570, 33985474184970, 250419761106900, 1853107999454250, 13765951702923420, 102618937160787060, 767411273728449480
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OFFSET
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0,2
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COMMENTS
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a(n) is also the number of n X 2 Young tableaux with (possibly) vertical walls. The entries in cells that are separated by such a wall do not have to obey any order constraints. See Banderier, Wallner 2021 and Banderier et al. 2018.
a(n) is also the number of binary trees with n vertices and marked leaves, where at least 1 leaf has to be marked. Banderier, Wallner 2021 give a bijection to n X 2 Young tableaux with vertical walls. (End)
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LINKS
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FORMULA
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D-finite with recurrence: a(n) = 2*(2*n-1)*(3*n*a(n-1)-4*(2*n-3)*a(n-2))/((n+1)*n). - Georg Fischer, Jun 06 2021
G.f.: A(x) = (sqrt(1-4*x) - sqrt(1-8*x))/(2*x).
G.f.: G(0)/(2*x) where G(k) = 1 - 2^k/(1 - 2*x*(2*k-1)/(2*x*(2*k-1) - 2^k*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012
a(n) = Sum_{k = 0..n} A046521(n,k)*Catalan(k).
G.f.: A(x) = 1/sqrt(1 - 4*x)*c(x/(1 - 4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Inversely, c(x) = 1/sqrt(1 + 4*x)*A(x/(1 + 4*x)).
Series reversion of x*A(x) = x*(1 - 3*x + 4*x^2*c(-2*x^2)). (End)
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MATHEMATICA
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Table[CatalanNumber[n]*(2^(n+1)-1), {n, 0, 20}] (* Harvey P. Dale, Oct 20 2014 *)
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PROG
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(PARI) a(n) = (2^(n + 1) - 1)*binomial(2*n, n)/(n + 1); \\ Harry J. Smith, Aug 16 2009
(Sage)
return (8^(n+1)-4^(n+1))*factorial(n-1/2)/(4*sqrt(pi)*factorial(n+1))
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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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EXTENSIONS
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STATUS
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approved
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