OFFSET
0,5
COMMENTS
Binomial transform is A098538.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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)
MATHEMATICA
CoefficientList[Series[(1+x)^(1/3)/(1+x-18*x^4)^(1/3), {x, 0, 50}], x] (* G. C. Greubel, Jan 17 2018 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Sep 13 2004
STATUS
approved