%I #50 Dec 02 2022 07:04:34
%S 1,5,30,198,1386,10098,75735,580635,4528953,35819901,286559208,
%T 2314516680,18846778680,154543585176,1274984577702,10574872085646,
%U 88123934047050,737458184920050,6194648753328420,52212039492339540,441429061162507020,3742550735942994300
%N Related to triple factorial numbers 2*A034000(n+1).
%H Vincenzo Librandi, <a href="/A034164/b034164.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).
%F G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.
%F G.f.: 2F1( (1, 5/3); 3; 9 x ). - _Olivier GĂ©rard_, Feb 15 2011
%F D-finite with recurrence: (n+2)*a(n) -3*(3*n+2)*a(n-1) = 0. - _R. J. Mathar_, Oct 29 2012
%F a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - _Vaclav Kotesovec_, Feb 09 2014
%F Integral representation as the n-th moment of a positive function on (0,9), in Maple notation: a(n) = int(x^n*W(x), x=0..9), n=0,1,..., 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 Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - _Amiram Eldar_, Dec 02 2022
%p 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 CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]
%t 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 CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)
%o (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 (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 (Sage)
%o def A034164_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()
%o A034164_list(30) # _G. C. Greubel_, Sep 17 2019
%Y Cf. A004990, A025748, A034000, A185047.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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