A004989 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).
1, -6, -9, -36, -189, -1134, -7371, -50544, -360126, -2640924, -19806930, -151252920, -1172210130, -9197341020, -72921775230, -583374201840, -4703454502335, -38180983607190, -311811366125385, -2560135427134740, -21121117273861605, -175003543126281870, -1455711290550435555
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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GAP
List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k-2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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Magma
[1] cat [3^n*(&*[3*k-2: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
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Maple
a:= n-> (3^n/n!)*product(3*k-2, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019
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Mathematica
Table[9^n*Pochhammer[-2/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *) -
PARI
a(n)=if(n<0,0,prod(k=0,n-1,3*k-2)*3^n/n!)
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Sage
[9^n*rising_factorial(-2/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019
Formula
a(n) ~ -(2/3)*Gamma(1/3)^-1*n^(-5/3)*3^(2*n)*(1 + (5/9)*n^-1 + ...).
G.f.: (1-9*x)^(2/3).
D-finite with recurrence: n*a(n) +3*(-3*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 99/128 - 5*sqrt(3)*Pi/512 - 15*log(3)/512. - Amiram Eldar, Dec 02 2022
From Peter Bala, Oct 14 2024: (Start)
a(n) = -1/(sqrt(3)*Pi) * 9^n * Gamma(2/3)*Gamma(n-2/3)/Gamma(n+1).
For n >= 1, a(n) = - Integral_{x = 0..9} x^n * w(x) dx, where w(x) = sqrt(3)/(2*Pi) * x^(1/3)*(9 - x)^(2/3)/x^2. (End)