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

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

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

Cf. A360560

Sequence in context: A267031 A289163 A092718 * A016131 A027310 A287826

Adjacent sequences: A090760 A090761 A090762 * A090764 A090765 A090766

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