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A383935 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(1/3).

%I A383935 #22 Aug 29 2025 03:18:48
%S A383935 1,3,12,54,261,1323,6930,37152,202554,1118286,6233760,35014356,
%T A383935 197881866,1123990182,6411554028,36705925656,210797967321,
%U A383935 1213895891835,7007131607220,40534622188830,234931402041525,1363961443750155,7931187074571930,46183636475060760
%N A383935 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(1/3).
%H A383935 Vincenzo Librandi, <a href="/A383935/b383935.txt">Table of n, a(n) for n = 0..1000</a>
%F A383935 G.f.: B(x)^(1/3), where B(x) is the g.f. of A016137.
%F A383935 a(n) = (-3)^n * Sum_{k=0..n} 2^k * binomial(-1/3,k) * binomial(-1/3,n-k).
%F A383935 a(n) ~ 2^(n + 1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Aug 18 2025
%F A383935 D-finite with recurrence: (-9*n-3)*a(n)+(18*n-6)*a(n-1)+(n+1)*a(n+1) = 0. - _Georg Fischer_, Aug 29 2025
%t A383935 CoefficientList[Series[1/((1-3*x)*(1-6*x))^(1/3),{x,0,33}],x] (* _Vincenzo Librandi_, Aug 28 2025 *)
%o A383935 (PARI) a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-1/3, k)*binomial(-1/3, n-k));
%o A383935 (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-3*x) * (1-6*x) )^(1/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 28 2025
%Y A383935 Cf. A016137, A383937.
%Y A383935 Cf. A383627.
%K A383935 nonn,easy,changed
%O A383935 0,2
%A A383935 _Seiichi Manyama_, Aug 18 2025