OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
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
EXAMPLE
a(1) = 4!/(2!*1!) = 24/2 = 12.
MAPLE
a:=n-> binomial(3*n+1, 2*n)*(n+1): seq(a(n), n=0..20); # Zerinvary Lajos, Jul 31 2006
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)excite.com), Feb 11 2004
EXTENSIONS
New definition from Vladeta Jovovic, Feb 12 2004
STATUS
approved