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%I #41 Feb 06 2024 09:02:38
%S 1,12,105,840,6435,48048,352716,2558160,18386775,131231100,931395465,
%T 6580248480,46312074900,324897017760,2272989850440,15863901576864,
%U 110487596768703,768095592509700,5330949171823275,36945070220658600,255702514854135195,1767643865751234240
%N a(n) = (3*n+1)!/((2*n)! * n!).
%H G. C. Greubel, <a href="/A090816/b090816.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 1/(Integral_{x=0..1} (x^2 - x^3)^n dx).
%F G.f.: (((8 + 27*z)*(1/(4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(1/3) + 1/(4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(1/3)) - 3*i*sqrt(3)*sqrt(4 - 27*z)*sqrt(z)*(1/(4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(1/3) - 1/(4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(1/3)))*8^(1/3))/(2*(4 - 27*z)^(3/2)), where i is the imaginary unit. - _Karol A. Penson_, Feb 06 2024
%e a(1) = 4!/(2!*1!) = 24/2 = 12.
%p a:=n-> binomial(3*n+1,2*n)*(n+1): seq(a(n), n=0..20); # _Zerinvary Lajos_, Jul 31 2006
%t f[n_] := 1/Integrate[(x^2 - x^3)^n, {x, 0, 1}]; Table[ f[n], {n, 0, 19}] (* _Robert G. Wilson v_, Feb 18 2004 *)
%t Table[1/Beta[2*n+1,n+1], {n,0,20}] (* _G. C. Greubel_, Feb 03 2019 *)
%o (PARI) a(n)=if(n<0,0,(3*n+1)!/(2*n)!/n!) /* _Michael Somos_, Feb 14 2004 */
%o (PARI) a(n)=if(n<0,0,1/subst(intformal((x^2-x^3)^n),x,1)) /* _Michael Somos_, Feb 14 2004 */
%o (Magma) [Factorial(3*n+1)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // _G. C. Greubel_, Feb 03 2019
%o (Sage) [1/beta(2*n+1,n+1) for n in range(20)] # _G. C. Greubel_, Feb 03 2019
%Y Cf. A045721, A002457, A090957, A090969.
%Y Halfdiagonal of triangle A003506.
%Y Equals A005809 * A016777.
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)excite.com), Feb 11 2004
%E New definition from _Vladeta Jovovic_, Feb 12 2004
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