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A115344 Numerators of asymptotic expansion of first root of Ziegler's cubic in an imaginary quadratic field.

%I A115344 #41 Dec 13 2024 09:40:43
%S A115344 -1,1,-1,3,-6,18,-45,136,-378,1156,-3405,10549,-32175,100915,-314834,
%T A115344 998323,-3163683,10127020,-32462265,104751043,-338742887,1100559573,
%U A115344 -3583933846,11711868458,-38358103030,125974533997,-414566089320,1367353737806,-4518185596293
%N A115344 Numerators of asymptotic expansion of first root of Ziegler's cubic in an imaginary quadratic field.
%H A115344 Vaclav Kotesovec, <a href="/A115344/b115344.txt">Table of n, a(n) for n = 0..500</a>
%H A115344 Volker Ziegler, <a href="http://finanz.math.tu-graz.ac.at/~ziegler/Papers/CubicsImaginary.pdf">On a family of cubics over imaginary quadratic fields</a>, Periodica Mathematica Hungarica, Vol. 51 (2) (2005), pp. 109-130, DOI: 10.1007/s10998-005-0032-6.
%F A115344 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A115344 (1) x = -(1+x)*A(x) - A(x)^2 + x*A(x)^3. - _Paul D. Hanna_, May 30 2014
%F A115344 (2) x = -A(x)*(1 + A(x)) / (1 + A(x) - A(x)^3). - _Paul D. Hanna_, May 31 2014
%F A115344 (3) A(x) = -x/Series_Reversion(x*(1 - Series_Reversion(x/(1 - 2*x + 3*x^2 - x^3)))). - _Paul D. Hanna_, May 31 2014
%F A115344 Recurrence: n*(n+1)*(28*n^2 - 94*n + 51)*a(n) = -4*n*(14*n^3 - 54*n^2 + 73*n - 48)*a(n-1) + (n-3)*(140*n^3 - 330*n^2 + 19*n + 216)*a(n-2) + 6*(n-3)*(28*n^3 - 108*n^2 + 57*n + 118)*a(n-3) + 23*(n-4)*(n-3)*(28*n^2 - 38*n - 15)*a(n-4). - _Vaclav Kotesovec_, May 30 2014
%F A115344 a(n) ~ (-1)^(n+1) * sqrt(s*(s-1)/(3*r*s-1)) / (2*sqrt(Pi) * n^(3/2)* r^n), where r = 2/(1+sqrt(13+16*sqrt(2))) = 0.2869905464691794898..., s = 1/2 + 1/sqrt(2) + 1/2*sqrt(2*sqrt(2)-1) = 1.88320350591352586... . - _Vaclav Kotesovec_, May 30 2014
%F A115344 a(n) = Sum_{k=0..n}(binomial(n,k)*Sum_{i=0..n-k-1}(2^(k-i)*binomial(k,i)*(-1)^(i+k)*binomial(2*n-i-2*k-2,n-k-1)))/n, n>0, a(0)=-1. - _Vladimir Kruchinin_, Mar 15 2016
%F A115344 a(n) = (1/n) * Sum_{k=0..n} (-1)^(n-k-1) * binomial(n,k) * binomial(2*n-3*k,n-k-1) for n > 0. - _Seiichi Manyama_, Dec 13 2024
%e A115344 -1 + 1/t - 1/t^2 + 3/t^3 - 6/t^4 + 18/t^5 - 45/t^6 + 136/t^7 - 378/t^8...
%t A115344 nmax=30; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=-1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[-(1+x)*AGF-AGF^2+x*AGF^3-x,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{-1,aa}] (* _Vaclav Kotesovec_, May 30 2014 *)
%t A115344 CoefficientList[-x/InverseSeries[x*(1-InverseSeries[Series[x/(1-2*x+3*x^2-x^3),{x,0,20}],x]),x],x] (* _Vaclav Kotesovec_, May 31 2014 after _Paul D. Hanna_ *)
%o A115344 (PARI) {a(n)=polcoeff(-x/serreverse(x*(1-serreverse(x/(1 - 2*x + 3*x^2 - x^3 +x*O(x^n))))), n)}
%o A115344 for(n=0,30,print1(a(n),", ")); \\ _Paul D. Hanna_, May 31 2014
%o A115344 (PARI) a(n) = if(n==0, -1, sum(k=0, n, (-1)^(n-k-1)*binomial(n, k)*binomial(2*n-3*k, n-k-1))/n); \\ _Seiichi Manyama_, Dec 13 2024
%o A115344 (Maxima)
%o A115344 a(n):=if n=0 then -1 else sum(binomial(n,k)*sum(2^(k-i)*binomial(k,i)*(-1)^(i+k)*binomial(2*n-i-2*k-2,n-k-1),i,0,n-k-1),k,0,n)/n; /* _Vladimir Kruchinin_, Mar 15 2016 */
%K A115344 sign
%O A115344 0,4
%A A115344 _Jonathan Vos Post_, Mar 06 2006
%E A115344 More terms from _Vaclav Kotesovec_, May 30 2014