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%I #16 Feb 10 2019 20:04:23
%S 1,2,10,59,385,2672,19336,144218,1100530,8549888,67386652,537437222,
%T 4328934754,35162809688,287688325672,2368563539171,19608128003473,
%U 163116600371846,1362822870625762,11430476370562259
%N Expansion of (1-(1-9*x)^(1/3))/(4-(1-9*x)^(1/3)).
%H Vincenzo Librandi, <a href="/A202482/b202482.txt">Table of n, a(n) for n = 1..200</a>
%F a(n):=1/n*sum(i=1..n, i*sum(k=0..n-i, binomial(k,n-k-i)*3^(k)*(-1)^(n-k+1)*binomial(n+k-1,n-1))).
%F Recurrence: 7*(n-1)*n*a(n) = (n-1)*(125*n - 252)*a(n-1) - 9*(61*n^2 - 309*n + 388)*a(n-2) - 9*(3*n-8)*(3*n-7)*a(n-3). - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ 9^n/(16*Gamma(2/3)*n^(4/3)). - _Vaclav Kotesovec_, Oct 20 2012
%t CoefficientList[Series[(1/x) (1- (1 - 9 x)^(1/3)) / (4 - (1 - 9 x)^(1/3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%t Module[{c=Surd[1-9x,3]},Rest[CoefficientList[Series[(1-c)/(4-c),{x,0,20}],x]]] (* _Harvey P. Dale_, Feb 10 2019 *)
%o (Maxima)
%o a(n):=sum(i*sum(binomial(k,n-k-i)*3^(k)*(-1)^(n-k+1)*binomial(n+k-1,n-1),k,0,n-i),i,1,n)/n;
%K nonn
%O 1,2
%A _Vladimir Kruchinin_, Dec 20 2011
%E Typo in Mathematica code fixed by _Vincenzo Librandi_, Jun 04 2013
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