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A383605 Expansion of 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3).

%I A383605 #17 May 06 2025 11:50:42
%S A383605 1,1,7,13,64,160,661,1927,7288,23044,83413,275479,976198,3301462,
%T A383605 11584861,39703783,138747637,479200129,1672353256,5803085008,
%U A383605 20251472416,70486033288,246114881956,858397066324,2999541427177,10477699520329,36642516789607,128146441442989
%N A383605 Expansion of 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3).
%H A383605 Vincenzo Librandi, <a href="/A383605/b383605.txt">Table of n, a(n) for n = 0..500</a>
%F A383605 a(n) = Sum_{k=0..floor(n/2)} (-9)^k * binomial(-2/3,k) * binomial(n-k,k).
%F A383605 a(n) ~ Gamma(1/3) * (1 + sqrt(37))^(n + 4/3) / (Pi * 3^(1/6) * 37^(1/3) * n^(1/3) * 2^(n + 7/3)). - _Vaclav Kotesovec_, May 02 2025
%t A383605 CoefficientList[Series[1/((1-x)*(1-x-9*x^2)^2)^(1/3),{x,0,27}],x] (* _Stefano Spezia_, May 02 2025 *)
%t A383605 Table[Sum[(-9)^k*Binomial[-2/3,k]*Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,35}] (* _Vincenzo Librandi_, May 06 2025 *)
%o A383605 (PARI) a(n) = sum(k=0, n\2, (-9)^k*binomial(-2/3, k)*binomial(n-k, k));
%o A383605 (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3))); // _Vincenzo Librandi_, May 06 2025
%Y A383605 Cf. A383601, A383606.
%K A383605 nonn
%O A383605 0,3
%A A383605 _Seiichi Manyama_, May 01 2025