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A024485
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a(n) = (2/(3*n-1))*binomial(3*n,n).
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2
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-2, 3, 6, 21, 90, 429, 2184, 11628, 63954, 360525, 2072070, 12096045, 71524440, 427496076, 2578547760, 15675792072, 95951017602, 590842763469, 3657598059570, 22749427475775, 142096423925610, 890949529108485, 5605635937900320, 35380499289211440, 223951032734902200
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OFFSET
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0,1
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COMMENTS
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For n >= 1, a(n) is the number of lattice paths from (0,0) to (2n,n) using only the steps (1,0) and (0,1) and which do not touch the line y = x/2 except at the path's endpoints. - Lucas A. Brown, Aug 21 2020
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LINKS
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FORMULA
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D-finite with recurrence: 2*n*(2*n-1)*a(n) -3*(3*n-2)*(3*n-4)*a(n-1)=0. - R. J. Mathar, Apr 27 2020
G.f.: - (sqrt(1-27*z/4)+i*sqrt(27*z/4))^(2/3) - (sqrt(1-27*z/4)-i*sqrt(27*z/4))^(2/3), where i = sqrt(-1).
(G.f.)^3 = G satisfies the cubic equation:
-(27*z - 2)^3 + 3*(27*z + 1)*(27*z - 5)*G + 3*(-27*z+2)*G^2 + G^3 = 0.
a(n) = Integral_{x=0..27/4} x^n*W(x) dx, for n>=1, where
W(x) = -(3^(1/6)*(9+sqrt(3)*sqrt(27-4*x))^(1/3))*(-27*(2^(1/3))*(3^(1/6)) + 3*2^(1/3)*3^(2/3)*sqrt(27-4*x)-2*(9+sqrt(3)*sqrt(27-4*x))^(1/3)*sqrt(27-4*x)*x^(1/3) + 4*2^(1/3)*3^(1/6)*x)/(4*2^(2/3)*Pi*sqrt(27-4*x)*x^(5/3)), for x in (0, 27/4). For n=0, Integral_{x=0..27/4} W(x) dx diverges and is not suited to reproduce a(0).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, and for x > 0 is monotonically decreasing to zero at x = 27/4. At x = 27/4 the first derivative of W(x) is infinite. (End)
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MAPLE
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[seq((2/(3*n-1))*binomial(3*n, n), n=0..40)];
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MATHEMATICA
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Table[2/(3n-1) Binomial[3n, n], {n, 0, 20}] (* Harvey P. Dale, Nov 21 2015 *)
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PROG
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(PARI) a(n) = (2/(3*n-1))*binomial(3*n, n); \\ Michel Marcus, May 10 2020
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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