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A067336
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a(0)=1, a(1)=2, a(n) = a(n-1)*9/2 - Catalan(n-1) where Catalan(n) = binomial(2n,n)/(n+1) = A000108(n).
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8
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1, 2, 8, 34, 148, 652, 2892, 12882, 57540, 257500, 1153888, 5175700, 23231864, 104335376, 468766292, 2106773874, 9470787588, 42583186476, 191494694352, 861248485884, 3873850923288, 17425765034376, 78391476387672
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OFFSET
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0,2
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COMMENTS
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Note that while a(n) is even (for n > 0), it is a multiple of 4 except when n = 2^m-1, i.e., when Catalan(n) is odd.
Result of applying the Riordan matrix ((1+sqrt(1-4x))/2, (1-sqrt(1-4x))/2) (inverse of (1/(1-x), x(1-x)) to 3^n. - Paul Barry, Mar 12 2005
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LINKS
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FORMULA
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G.f.: (1+sqrt(1-4x))/(3*sqrt(1-4x)-1). - Paul Barry, Mar 12 2005
G.f.: (1-x*c(x))/(1-3*x*c(x)), where c(x) is the g.f. of A000108. - Paul Barry, Mar 15 2010
Conjecture: 2*n*a(n) + (-17*n+12)*a(n-1) + 18*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
G.f.: 1 + 2*x/(Q(0)-3*x), where Q(k) = 2*x + (k+1)/(2*k+1) - 2*x*(k+1)/(2*k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
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EXAMPLE
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a(2) = 2*9/2 - 1 = 8;
a(3) = 8*9/2 - 2 = 34;
a(4) = 34*9/2 - 5 = 148;
a(5) = 148*9/2 - 14 = 652.
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MATHEMATICA
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CoefficientList[Series[(1+Sqrt[1-4*x])/(3*Sqrt[1-4*x]-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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