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A386721 Expansion of e.g.f. exp(x/(1 - 9*x^3)^(1/3)).

%I A386721 #41 Sep 03 2025 10:50:03
%S A386721 1,1,1,1,73,361,1081,93241,912241,4907953,476295121,7244922961,
%T A386721 58360393081,6211842488281,130899060524233,1435239754046281,
%U A386721 164948740478252641,4498516738183799521,63300797606830713121,7772118657831401082913,262261735708117281036841
%N A386721 Expansion of e.g.f. exp(x/(1 - 9*x^3)^(1/3)).
%H A386721 Vincenzo Librandi, <a href="/A386721/b386721.txt">Table of n, a(n) for n = 0..300</a>
%H A386721 Vaclav Kotesovec, <a href="/A386721/a386721.jpg">Graph - the asymptotic ratio (20000 terms)</a>
%F A386721 a(n) = n! * Sum_{k=0..floor(n/3)} 9^k * binomial(n/3-1,k)/(n-3*k)!.
%F A386721 a(n) == 1 mod 72.
%F A386721 From _Vaclav Kotesovec_, Sep 03 2025: (Start)
%F A386721 a(n) = (36*n^3 - 324*n^2 + 1008*n - 1079)*a(n-3) - 162*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 2916*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - 6561*(n-12)*(n-11)*(n-10)*(n-9)^2*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*a(n-12).
%F A386721 a(n) ~ 3^(2*n/3 - 1/4) * exp(4*3^(-3/2)*n^(1/4) - n) * n^(n - 3/8) / 2. (End)
%t A386721 nmax = 20; CoefficientList[Series[E^(x/(1 - 9*x^3)^(1/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 03 2025 *)
%o A386721 (PARI) a(n) = n!*sum(k=0, n\3, 9^k*binomial(n/3-1, k)/(n-3*k)!);
%o A386721 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1 - 9*x^3)^(1/3)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A386721 Cf. A189054, A371458, A373517.
%K A386721 nonn,new
%O A386721 0,5
%A A386721 _Seiichi Manyama_, Sep 03 2025