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%I #34 Feb 12 2023 10:00:09
%S 1,10,84,660,5005,37128,271320,1961256,14060475,100150050,709634640,
%T 5006710800,35197176924,246681069040,1724337127920,12025860872784,
%U 83702724824775,581558091471630,4034231805704100,27945630038703300
%N a(n) = (3*n+3)!/(3*n!*(2*n+2)!).
%H Alois P. Heinz, <a href="/A090763/b090763.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = 1/(Integral_{x=0..1} (x^(2/3)-x)^n dx).
%F a(n) = 1/(Integral_{x=0..1} (x-x^1.5)^n dx).
%F a(n) = 1/(2*Beta(2n,n)). [_Enrique Pérez Herrero_, May 17 2009]
%F a(1) = 1; a(n) = a(n-1)*2*binomial(3n,3)/binomial(2n,3). - _Enrique Pérez Herrero_, May 19 2009
%F a(n) = (1/2)*Sum{j=1,n}(j*binomial(2n,j)*binomial(n,j)). - _Enrique Pérez Herrero_, May 22 2009
%F a(n) = (n+1)*A025174(n+1). - _R. J. Mathar_, Jun 21 2009
%F G.f.: Hypergeometric2F1(4/3, 5/3, 3/2, 27*x/4). - _Stefano Spezia_, Oct 18 2019
%F 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
%e E.g. a(3)=660.
%p 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
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1,
%p 3*(3*n+1)*(3*n+2)*a(n-1)/(2*n*(2*n+1)))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 01 2014
%t 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 *)
%t a[n_] := 1/(2*Beta[2n, n]) (* _Enrique Pérez Herrero_, May 17 2009 *)
%t a[n_]:=1/2*Sum[j*Binomial[2 n, j]*Binomial[n, j], {j, 1, n}] (* _Enrique Pérez Herrero_, May 22 2009 *)
%o (Sage) [binomial(3*n,n)*n/3 for n in range(1,21)] # _Zerinvary Lajos_, May 17 2009
%Y Cf. A360560
%K nonn
%O 0,2
%A Al Hakanson (hawkuu(AT)excite.com), Feb 15 2004
%E More terms from _Robert G. Wilson v_, Feb 18 2004
%E Simpler description from _Vladeta Jovovic_, Feb 22 2004
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