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

-1, 1, -1, 3, -6, 18, -45, 136, -378, 1156, -3405, 10549, -32175, 100915, -314834, 998323, -3163683, 10127020, -32462265, 104751043, -338742887, 1100559573, -3583933846, 11711868458, -38358103030, 125974533997, -414566089320, 1367353737806, -4518185596293

Offset: 0

Author

Jonathan Vos Post, Mar 06 2006

Keywords

• sign

Examples

			-1 + 1/t - 1/t^2 + 3/t^3 - 6/t^4 + 18/t^5 - 45/t^6 + 136/t^7 - 378/t^8...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500
• Volker Ziegler, On a family of cubics over imaginary quadratic fields, Periodica Mathematica Hungarica, Vol. 51 (2) (2005), pp. 109-130, DOI: 10.1007/s10998-005-0032-6.

Programs

Mathematica

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 *)
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 *)

Maxima

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 */

PARI

{a(n)=polcoeff(-x/serreverse(x*(1-serreverse(x/(1 - 2*x + 3*x^2 - x^3 +x*O(x^n))))), n)}
for(n=0,30,print1(a(n),", ")); \\ Paul D. Hanna, May 31 2014

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) x = -(1+x)*A(x) - A(x)^2 + x*A(x)^3. - Paul D. Hanna, May 30 2014

(2) x = -A(x)*(1 + A(x)) / (1 + A(x) - A(x)^3). - Paul D. Hanna, May 31 2014

(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

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

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

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

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

Extensions

More terms from Vaclav Kotesovec, May 30 2014