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A090763
a(n) = (3*n+3)!/(3*n!*(2*n+2)!).
5
1, 10, 84, 660, 5005, 37128, 271320, 1961256, 14060475, 100150050, 709634640, 5006710800, 35197176924, 246681069040, 1724337127920, 12025860872784, 83702724824775, 581558091471630, 4034231805704100, 27945630038703300
OFFSET
0,2
LINKS
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
EXAMPLE
E.g. a(3)=660.
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
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 *)
PROG
(Sage) [binomial(3*n, n)*n/3 for n in range(1, 21)] # Zerinvary Lajos, May 17 2009
CROSSREFS
Sequence in context: A267031 A289163 A092718 * A016131 A027310 A287826
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)excite.com), Feb 15 2004
EXTENSIONS
More terms from Robert G. Wilson v, Feb 18 2004
Simpler description from Vladeta Jovovic, Feb 22 2004
STATUS
approved