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

1, 0, 0, 0, 6, -6, 6, -6, 78, -150, 222, -294, 1374, -3462, 6558, -10662, 30894, -82374, 180222, -339558, 811374, -2082534, 4875774, -10149702, 22872750, -55797126, 133232766, -294821286, 660771438, -1558556070, 3711070590

Offset: 0

Author

Paul Barry, Sep 13 2004

Keywords

• easy
• sign

Comments

Binomial transform is A098538.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A098535.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q,30); Coefficients(R!((1+x)^(1/3)/(1+x-18*x^4)^(1/3))); // G. C. Greubel, Jan 17 2018

Mathematica

CoefficientList[Series[(1+x)^(1/3)/(1+x-18*x^4)^(1/3), {x, 0, 50}], x] (* G. C. Greubel, Jan 17 2018 *)

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 */

PARI

x='x+O('x^30); Vec((1+x)^(1/3)/(1+x-18*x^4)^(1/3)) \\ G. C. Greubel, Jan 17 2018

Formula

From Vladimir Kruchinin, Sep 06 2010: (Start)

a(n) = Sum(b(j)*c(n-j,j,0,n), where:

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); (End)