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%I #24 Dec 22 2022 08:40:24
%S 1,6,42,315,2457,19656,160056,1320462,11003850,92432340,781473420,
%T 6642524070,56716936290,486145168200,4180848446520,36059817851235,
%U 311811366125385,2702365173086670,23467908082068450,204170800313995515,1779202688450532345,15527587099204645920
%N Related to triple factorial numbers A007559(n+1).
%C Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - _Peter Bala_, Dec 23 2019
%H Michael De Vlieger, <a href="/A034171/b034171.txt">Table of n, a(n) for n = 0..1050</a>
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.
%F G.f.: (-1+(1-9*x)^(-1/3))/(3*x).
%F a(n)= A035529(n+1, 1) (first column of triangle).
%F Convolution of A004987(n) with A025748(n+1), n >= 0.
%F D-finite with recurrence: (n+1)*a(n) +3*(-3*n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 28 2020
%F G.f.: (1F0(1/3;;9*x)-1)/(3*x). - _R. J. Mathar_, Jan 28 2020
%F Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - _Amiram Eldar_, Dec 22 2022
%t CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* _Michael De Vlieger_, Oct 13 2019 *)
%Y Cf. A007559, A025748, A034164, A035529.
%Y Cf. A298799, A004981, A004982.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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