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A091541 Four times triple factorials (3*n-2)!!! with leading 1 added.

%I A091541 #17 Aug 30 2025 02:43:34
%S A091541 1,4,4,16,112,1120,14560,232960,4426240,97377280,2434432000,
%T A091541 68164096000,2113086976000,71844957184000,2658263415808000,
%U A091541 106330536632320000,4572213075189760000,210321801458728960000,10305768271477719040000,535899950116841390080000,29474497256426276454400000
%N A091541 Four times triple factorials (3*n-2)!!! with leading 1 added.
%C A091541 The exponential (or binomial) convolution of a(n) with A051606(n) gives A091540.
%H A091541 G. C. Greubel, <a href="/A091541/b091541.txt">Table of n, a(n) for n = 0..381</a>
%F A091541 a(0) = 1, a(n) = 4*(3*n-2)!!! = 4*A007559(n-1), n>=1.
%F A091541 E.g.f. 3-2*(1-3*x)^(2/3).
%F A091541 E.g.f. for a(n+1)/4 = A007559(n), n>=0: (1-3*x)^(-1/3).
%F A091541 G.f.: 3-G(0), where G(k)= 1 + 1/(1 - x*(3*k-2)/(x*(3*k-2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 11 2013
%F A091541 From _Amiram Eldar_, Aug 30 2025: (Start)
%F A091541 a(n) ~ 4 * sqrt(2*Pi) * 3^(n-1) * n^(n-7/6) / (Gamma(1/3) * exp(n)).
%F A091541 Sum_{n>=0} 1/a(n) = (5 + (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3))) / 4. (End)
%t A091541 With[{nmax = 50}, CoefficientList[Series[3 - 2*(1 - 3*x)^(2/3), {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Aug 15 2018 *)
%o A091541 (PARI) my(x='x+O('x^50)); Vec(serlaplace(3 - 2*(1 - 3*x)^(2/3))) \\ _G. C. Greubel_, Aug 15 2018
%o A091541 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(3 - 2*(1 - 3*x)^(2/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 15 2018
%Y A091541 Cf. A007559, A051606, A091540.
%K A091541 nonn,easy,changed
%O A091541 0,2
%A A091541 _Wolfdieter Lang_, Feb 13 2004