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 A090763 a(n) = (3*n+3)!/(3*n!*(2*n+2)!). 4
 1, 10, 84, 660, 5005, 37128, 271320, 1961256, 14060475, 100150050, 709634640, 5006710800, 35197176924, 246681069040, 1724337127920, 12025860872784, 83702724824775, 581558091471630, 4034231805704100, 27945630038703300 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 27 20:41 EDT 2023. Contains 365714 sequences. (Running on oeis4.)