A090763 a(n) = (3*n+3)!/(3*n!*(2*n+2)!).
1, 10, 84, 660, 5005, 37128, 271320, 1961256, 14060475, 100150050, 709634640, 5006710800, 35197176924, 246681069040, 1724337127920, 12025860872784, 83702724824775, 581558091471630, 4034231805704100, 27945630038703300
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..500
Programs
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Maple
a:= n->sum(j*binomial(n+2, j)*binomial(2*(n+1), j)/6, j=0..n+2): seq(a(n), n=0..21); # Zerinvary Lajos, Jul 31 2006 # second Maple program: a:= proc(n) option remember; `if`(n=0, 1, 3*(3*n+1)*(3*n+2)*a(n-1)/(2*n*(2*n+1))) end: seq(a(n), n=0..30); # Alois P. Heinz, Feb 01 2014
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Mathematica
a[n_] := 1/Integrate[(x^(2/3) - x)^n, {x, 0, 1}]; Table[ a[n], {n, 0, 19}] (* Robert G. Wilson v, Feb 18 2004 *) a[n_] := 1/(2*Beta[2n, n]) (* Enrique Pérez Herrero, May 17 2009 *) a[n_] : =1/2*Sum[j*Binomial[2 n, j]*Binomial[n, j], {j, 1, n}] (* Enrique Pérez Herrero, May 22 2009 *) -
Sage
[binomial(3*n,n)*n/3 for n in range(1,21)] # Zerinvary Lajos, May 17 2009
Formula
a(n) = 1/(Integral_{x=0..1} (x^(2/3)-x)^n dx).
a(n) = 1/(Integral_{x=0..1} (x-x^1.5)^n dx).
a(n) = 1/(2*Beta(2n,n)). - Enrique Pérez Herrero, May 17 2009
a(1) = 1; a(n) = a(n-1)*2*binomial(3n,3)/binomial(2n,3). - Enrique Pérez Herrero, May 19 2009
a(n) = (1/2)*Sum{j=1,n}(j*binomial(2n,j)*binomial(n,j)). - Enrique Pérez Herrero, May 22 2009
a(n) = (n+1)*A025174(n+1). - R. J. Mathar, Jun 21 2009
G.f.: Hypergeometric2F1(4/3, 5/3, 3/2, 27*x/4). - Stefano Spezia, Oct 18 2019
G.f.: (-(3*sqrt(4-27*x)*csc(arcsin((3*sqrt(3*x))/2)/3)^2)/((4*(4-27*x)^(3/2)))+(sqrt(3)*cot(arcsin((3*sqrt(3*x))/2)/3))/((4-27*x)*sqrt(x)*sqrt(4-27*x))). - Vladimir Kruchinin, Feb 12 2023
From Amiram Eldar, Dec 07 2024: (Start)
a(n) = (n+1) * A005809(n+1) / 3.
Sum_{n>=0} 1/a(n) = 3 * A210453. (End)
Extensions
More terms from Robert G. Wilson v, Feb 18 2004
Simpler description from Vladeta Jovovic, Feb 22 2004