OFFSET
0,2
COMMENTS
What is the behavior of this sequence? Does there exist a g.f.?
It would be nice to know the (more accurate) values of the following limits:
(1) The position of the first negative coefficient in (1+x+x^2-x^3)^n, divided by n, seems to reach a limit near 0.398...
(2) Limit a(n)^(1/n) seems to exist near 2.6637...
(3) Limit a(n+1)/a(n) does not seem to be unique; attractors seem to exist near 2.66...
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
EXAMPLE
Illustrate the coefficients in (1+x+x^2-x^3)^n by:
n=0: [1];
n=1: [1, 1, 1, -1];
n=2: [1, 2, 3, 0, -1, -2, 1];
n=3: [1, 3, 6, 4, 0, -6, -2, 0, 3, -1];
n=4: [1, 4, 10, 12, 7, -8, -12, -8, 7, 4, 2, -4, 1];
n=5: [1, 5, 15, 25, 25, 1, -25, -35, -5, 15, 21, -5, -5, -5, 5, -1];
n=6: [1, 6, 21, 44, 60, 36, -24, -84, -66, 0, 66, 36, -4, -36, 0, 4, 9, -6, 1];
n=7: [1, 7, 28, 70, 119, 119, 28, -132, -210, -126, 84, 168, 98, -70, -76, -28, 49, 7, 0, -14, 7, -1]; ...
This sequence gives the sums of the absolute values of the coefficients for n>=0.
MATHEMATICA
Table[Total[Abs[CoefficientList[Expand[(1+x+x^2-x^3)^n], x]]], {n, 0, 30}] (* Harvey P. Dale, Oct 12 2012 *)
PROG
(PARI) {a(n)=sum(k=0, 3*n, abs(polcoeff((1+x+x^2-x^3)^n, k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 25 2011
STATUS
approved