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A090816 a(n) = (3*n+1)!/((2*n)! * n!). 5
1, 12, 105, 840, 6435, 48048, 352716, 2558160, 18386775, 131231100, 931395465, 6580248480, 46312074900, 324897017760, 2272989850440, 15863901576864, 110487596768703, 768095592509700, 5330949171823275, 36945070220658600, 255702514854135195, 1767643865751234240 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
Halfdiagonal of triangle A003506.
Equals A005809 * A016777.
Sequence in context: A004321 A016223 A027142 * A244722 A144133 A089396
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)excite.com), Feb 11 2004
EXTENSIONS
New definition from Vladeta Jovovic, Feb 12 2004
STATUS
approved

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Last modified April 19 12:14 EDT 2024. Contains 371792 sequences. (Running on oeis4.)