A090816 a(n) = (3*n+1)!/((2*n)! * n!).
1, 12, 105, 840, 6435, 48048, 352716, 2558160, 18386775, 131231100, 931395465, 6580248480, 46312074900, 324897017760, 2272989850440, 15863901576864, 110487596768703, 768095592509700, 5330949171823275, 36945070220658600, 255702514854135195, 1767643865751234240
Offset: 0
Examples
a(1) = 4!/(2!*1!) = 24/2 = 12.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Necdet Batir, On certain series involving reciprocals of binomial coefficients, Journal of Classical Analysis, Vol. 2, No. 1 (2013), pp. 1-8.
Crossrefs
Programs
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Magma
[Factorial(3*n+1)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // G. C. Greubel, Feb 03 2019
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Maple
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
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 *) -
PARI
a(n)=if(n<0,0,(3*n+1)!/(2*n)!/n!) /* Michael Somos, Feb 14 2004 */
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PARI
a(n)=if(n<0,0,1/subst(intformal((x^2-x^3)^n),x,1)) /* Michael Somos, Feb 14 2004 */
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Sage
[1/beta(2*n+1,n+1) for n in range(20)] # G. C. Greubel, Feb 03 2019
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
a(n) = Sum_{k = 0..n} (-1)^(n+k) * (2*n + 2*k + 1)*binomial(2*n+k, k). This is the particular case m = 1 of the identity Sum_{k = 0..m*n} (-1)^k * (2*n + 2*k + 1) * binomial(2*n+k, k) = (-1)^(m*n) * (m*n + 1) * binomial((m+2)*n+1, 2*n). Cf. A002457 and A306290. - Peter Bala, Nov 02 2024
From Amiram Eldar, Dec 09 2024: (Start)
Sum_{n>=0} 1/a(n) = f(c) = 1.09422712102982285131..., where f(x) = (x*(x-1)/(3*x-1)) * ((3/2)*log(abs(x/(x-1))) + ((3*x-2)/sqrt(3*x^2-4*x)) * (arctan(x/sqrt(3*x^2-4*x)) + arctan((2-x)/sqrt(3*x^2-4*x)))), and c = 2/3 + (1/3)*((25+3*sqrt(69))/2)^(-1/3) + (1/3)*((25+3*sqrt(69))/2)^(1/3).
Sum_{n>=0} (-1)^n/a(n) = f(d) = 0.92513707957813718109..., where f(x) is defined above, and d = 2/3 - (1/3)*((29+3*sqrt(93))/2)^(-1/3) - (1/3)*((29+3*sqrt(93))/2)^(1/3).
Both formulas are from Batir (2013). (End)
Extensions
New definition from Vladeta Jovovic, Feb 12 2004