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A024485 a(n) = (2/(3*n-1))*binomial(3*n,n). 2
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
FORMULA
G.f.: 3*g-2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
a(n) = 2*A005809(n)/(3*n-1). - R. J. Mathar, Apr 27 2020
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
a(n) = A006013(n-1)/3 for n >= 1. - Lucas A. Brown, Aug 21 2020
From Karol A. Penson, Dec 18 2023: (Start)
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)
MAPLE
[seq((2/(3*n-1))*binomial(3*n, n), n=0..40)];
MATHEMATICA
Table[2/(3n-1) Binomial[3n, n], {n, 0, 20}] (* Harvey P. Dale, Nov 21 2015 *)
PROG
(PARI) a(n) = (2/(3*n-1))*binomial(3*n, n); \\ Michel Marcus, May 10 2020
CROSSREFS
Sequence in context: A025239 A127294 A012924 * A013155 A303224 A007501
KEYWORD
sign
AUTHOR
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, May 10 2020
STATUS
approved

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Last modified June 24 06:40 EDT 2024. Contains 373663 sequences. (Running on oeis4.)