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 A034164 #59 Aug 19 2025 04:23:17
%S A034164 1,5,30,198,1386,10098,75735,580635,4528953,35819901,286559208,
%T A034164 2314516680,18846778680,154543585176,1274984577702,10574872085646,
%U A034164 88123934047050,737458184920050,6194648753328420,52212039492339540,441429061162507020,3742550735942994300
%N A034164 Related to triple factorial numbers 2*A034000(n+1).
%H A034164 Vincenzo Librandi, <a href="/A034164/b034164.txt">Table of n, a(n) for n = 0..200</a>
%F A034164 a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).
%F A034164 G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.
%F A034164 G.f.: 2F1( (1, 5/3); 3; 9 x ). - _Olivier Gérard_, Feb 15 2011
%F A034164 D-finite with recurrence: (n+2)*a(n) - 3*(3*n+2)*a(n-1) = 0. - _R. J. Mathar_, Oct 29 2012
%F A034164 a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - _Vaclav Kotesovec_, Feb 09 2014
%F A034164 Integral representation as the n-th moment of a positive function on (0,9): a(n) = Integral_{x=0..9} x^n*W(x) dx, n >= 0, where W(x) = (1/18)*9^(1/3)*sqrt(3)*x^(2/3)*(1-x/9)^(1/3)/Pi. This representation is unique as W(x) is the solution of the Hausdorff moment problem. - _Karol A. Penson_, Nov 07 2015
%F A034164 Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - _Amiram Eldar_, Dec 02 2022
%F A034164 a(n) ~ 3^(2*n+1) * n^(-4/3) / Gamma(2/3). - _Amiram Eldar_, Aug 19 2025
%p A034164 seq(coeff(series((1-3*x-(1-9*x)^(1/3))/(3*x)^2, x, n+2), x, n), n = 0..32); # _G. C. Greubel_, Sep 17 2019
%t A034164 CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]
%t A034164 Table[FullSimplify[3^(2*n+1) * Gamma[n+5/3] / ((n+2) * Gamma[2/3] * Gamma[n+2])],{n,0,20}] (* _Vaclav Kotesovec_, Feb 09 2014 *)
%t A034164 CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)
%o A034164 (PARI) my(x='x+O('x^30)); Vec((1 -3*x -(1-9*x)^(1/3))/(3*x)^2) \\ _G. C. Greubel_, Sep 17 2019
%o A034164 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 -3*x -(1-9*x)^(1/3))/(3*x)^2 )); // _G. C. Greubel_, Sep 17 2019
%o A034164 (Sage)
%o A034164 def A034164_list(prec):
%o A034164 P.<x> = PowerSeriesRing(QQ, prec)
%o A034164 return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()
%o A034164 A034164_list(30) # _G. C. Greubel_, Sep 17 2019
%Y A034164 Cf. A004990, A025748, A034000, A073006, A185047.
%K A034164 nonn,easy
%O A034164 0,2
%A A034164 _Wolfdieter Lang_