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A098537 Expansion of (1+x)^(1/3)/(1+x-18*x^4)^(1/3).

%I A098537 #11 Sep 08 2022 08:45:15
%S A098537 1,0,0,0,6,-6,6,-6,78,-150,222,-294,1374,-3462,6558,-10662,30894,
%T A098537 -82374,180222,-339558,811374,-2082534,4875774,-10149702,22872750,
%U A098537 -55797126,133232766,-294821286,660771438,-1558556070,3711070590
%N A098537 Expansion of (1+x)^(1/3)/(1+x-18*x^4)^(1/3).
%C A098537 Binomial transform is A098538.
%H A098537 G. C. Greubel, <a href="/A098537/b098537.txt">Table of n, a(n) for n = 0..1000</a>
%F A098537 From _Vladimir Kruchinin_, Sep 06 2010: (Start)
%F A098537 a(n) = Sum(b(j)*c(n-j,j,0,n), where:
%F A098537 b(n) = if n=0 then 1 else Sum(Sum((if mod(n-4*k,3)=0 then binomial(k,(4*k-n)/3)*(-1)^((4*k-n)/3)*(18)^((n-k)/3) else 0)*(if k=m then (1/3)^k else m/k*(1/3)^k*Sum(binomial(i,k-m-i)*(-1/3)^(k-m-i)*binomial(i+k-1,k-1),i,1,k-m)),k,m,n),m,1,n),
%F A098537 c(n)=if n=0 then 1 else (-1)^(n+1)*if n=1 then (1/3)^n else 1/n*(1/3)^n * Sum(binomial(k,n-1-k)*(-1/3)^(n-1-k)*binomial(k+n-1,n-1),k,1,n-1); (End)
%t A098537 CoefficientList[Series[(1+x)^(1/3)/(1+x-18*x^4)^(1/3), {x, 0, 50}], x] (* _G. C. Greubel_, Jan 17 2018 *)
%o A098537 (Maxima) a(n):=sum(b(j)*c(n-j,j,0,n); b(n):=if n=0 then 1 else sum(sum((if mod(n-4*k,3)=0 then binomial(k,(4*k-n)/3)*(-1)^((4*k-n)/3)*(18)^((n-k)/3) else 0)*(if k=m then (1/3)^k else m/k*(1/3)^k*sum(binomial(i,k-m-i)*(-1/3)^(k-m-i)*binomial(i+k-1,k-1),i,1,k-m)),k,m,n),m,1,n); c(n):=if n=0 then 1 else (-1)^(n+1)*if n=1 then (1/3)^n else 1/n*(1/3)^n*sum(binomial(k,n-1-k)*(-1/3)^(n-1-k)*binomial(k+n-1,n-1),k,1,n-1);  /* _Vladimir Kruchinin_, Sep 06 2010 */
%o A098537 (PARI) x='x+O('x^30); Vec((1+x)^(1/3)/(1+x-18*x^4)^(1/3)) \\ _G. C. Greubel_, Jan 17 2018
%o A098537 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q,30); Coefficients(R!((1+x)^(1/3)/(1+x-18*x^4)^(1/3))); // _G. C. Greubel_, Jan 17 2018
%Y A098537 Cf. A098535.
%K A098537 easy,sign
%O A098537 0,5
%A A098537 _Paul Barry_, Sep 13 2004