This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A004990 #56 Apr 02 2024 11:44:42
%S A004990 1,-3,-9,-45,-270,-1782,-12474,-90882,-681615,-5225715,-40760577,
%T A004990 -322379109,-2579032872,-20830650120,-169621008120,-1390892266584,
%U A004990 -11474861199318,-95173848770814,-793115406423450,-6637123664280450,-55751838779955780
%N A004990 a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k - 1).
%H A004990 Seiichi Manyama, <a href="/A004990/b004990.txt">Table of n, a(n) for n = 0..1000</a>
%F A004990 a(n) = 9*A034164(n-2), n >= 2.
%F A004990 G.f.: (1 - 9*x)^(1/3).
%F A004990 a(n) ~ -1/3*Gamma(2/3)^-1*n^(-4/3)*3^(2*n)*{1 + 2/9*n^-1 + ...}.
%F A004990 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
%F A004990 D-finite with recurrence: n*a(n) +3*(-3*n+4)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F A004990 Sum_{n>=0} 1/a(n) = 9/16 - sqrt(3)*Pi/64 + 3*log(3)/64. - _Amiram Eldar_, Dec 02 2022
%F A004990 From _Peter Bala_, Mar 31 2024: (Start)
%F A004990 a(n) = (-9)^n*binomial(1/3, n).
%F A004990 E.g.f.: hypergeom([-1/3], [1], 9*x).
%F A004990 a(n) = (9^n)*Sum_{k = 0..2*n} (-1)^k*binomial(1/3, k)* binomial(1/3, 2*n - k).
%F A004990 (9^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
%F A004990 Sum_{k = 0..n} a(k)*a(n-k) = A004989.
%F A004990 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)
%p A004990 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
%t A004990 FullSimplify[Table[3^(2*n) * Gamma[n-1/3] / (n! * Gamma[-1/3]),{n,0,20}]] (* _Vaclav Kotesovec_, Dec 03 2014 *)
%o A004990 (PARI) for(n=0,30,print1( (3^n/n!)*prod(k=0,n-1,(3*k-1) ),","))
%o A004990 (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
%o A004990 (Sage) [9^n*rising_factorial(-1/3, n)/factorial(n) for n in (0..20)] # _G. C. Greubel_, Aug 22 2019
%o A004990 (GAP) List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k-1)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019
%Y A004990 Cf. A004988, A004989, A004991, A004992, A034164.
%K A004990 sign,easy
%O A004990 0,2
%A A004990 Joe Keane (jgk(AT)jgk.org)
%E A004990 More terms from _Jason Earls_, Dec 03 2001