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A090816
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a(n) = (3*n+1)!/((2*n)! * n!).
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6
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1, 12, 105, 840, 6435, 48048, 352716, 2558160, 18386775, 131231100, 931395465, 6580248480, 46312074900, 324897017760, 2272989850440, 15863901576864, 110487596768703, 768095592509700, 5330949171823275, 36945070220658600, 255702514854135195, 1767643865751234240
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 1/(Integral_{x=0..1} (x^2 - x^3)^n dx).
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
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EXAMPLE
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a(1) = 4!/(2!*1!) = 24/2 = 12.
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MAPLE
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a:=n-> binomial(3*n+1, 2*n)*(n+1): seq(a(n), n=0..20); # Zerinvary Lajos, Jul 31 2006
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MATHEMATICA
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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 *)
Table[1/Beta[2*n+1, n+1], {n, 0, 20}] (* G. C. Greubel, Feb 03 2019 *)
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PROG
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(PARI) a(n)=if(n<0, 0, (3*n+1)!/(2*n)!/n!) /* Michael Somos, Feb 14 2004 */
(PARI) a(n)=if(n<0, 0, 1/subst(intformal((x^2-x^3)^n), x, 1)) /* Michael Somos, Feb 14 2004 */
(Magma) [Factorial(3*n+1)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // G. C. Greubel, Feb 03 2019
(Sage) [1/beta(2*n+1, n+1) for n in range(20)] # G. C. Greubel, Feb 03 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Al Hakanson (hawkuu(AT)excite.com), Feb 11 2004
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EXTENSIONS
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STATUS
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approved
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