A004990 a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k - 1).
1, -3, -9, -45, -270, -1782, -12474, -90882, -681615, -5225715, -40760577, -322379109, -2579032872, -20830650120, -169621008120, -1390892266584, -11474861199318, -95173848770814, -793115406423450, -6637123664280450, -55751838779955780
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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GAP
List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k-1)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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Magma
[1] cat [3^n*(&*[3*k-1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 22 2019
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Maple
a:= n-> (3^n/n!)*mul(3*k-1, k=0..n-1): seq(a(n), n=0..20); # G. C. Greubel, Aug 22 2019
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Mathematica
FullSimplify[Table[3^(2*n) * Gamma[n-1/3] / (n! * Gamma[-1/3]),{n,0,20}]] (* Vaclav Kotesovec, Dec 03 2014 *) -
PARI
for(n=0,30,print1( (3^n/n!)*prod(k=0,n-1,(3*k-1) ),","))
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Sage
[9^n*rising_factorial(-1/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019
Formula
a(n) = 9*A034164(n-2), n >= 2.
G.f.: (1 - 9*x)^(1/3).
a(n) ~ -1/3*Gamma(2/3)^-1*n^(-4/3)*3^(2*n)*{1 + 2/9*n^-1 + ...}.
G.f.: 1 + 3*x/(G(0)-3*x) where G(k) = (1+9*x)*k + 1 - 3*x - 3*x*(k+1)*(3*k+2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 07 2012
D-finite with recurrence: n*a(n) +3*(-3*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 9/16 - sqrt(3)*Pi/64 + 3*log(3)/64. - Amiram Eldar, Dec 02 2022
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-9)^n*binomial(1/3, n).
E.g.f.: hypergeom([-1/3], [1], 9*x).
a(n) = (9^n)*Sum_{k = 0..2*n} (-1)^k*binomial(1/3, k)* binomial(1/3, 2*n - k).
(9^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = A004989.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = 18^n/(2*n)! * Product_{k = 1..n} (6*k - 5)*(3*k - 4). (End)
Extensions
More terms from Jason Earls, Dec 03 2001