A192210 a(n) = sum of unsigned coefficients in (1+x+x^2-x^3)^n.

1, 4, 10, 26, 80, 194, 504, 1442, 3710, 9536, 26842, 69014, 178704, 496602, 1316204, 3377206, 9242898, 24629944, 63304540, 172497622, 462822414, 1210912388, 3177522724, 8736822276, 22617998204, 59776061150, 163702751968, 433787373560

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

Cf. A192205, A084611, A084613, A084615, A084775, A084776, A084777, A084778, A084779, A084780.

Sequence in context: A130583 A240048 A105999 * A121494 A122744 A192879

Adjacent sequences: A192207 A192208 A192209 * A192211 A192212 A192213

Keyword

nonn

Author

Paul D. Hanna, Jun 25 2011

Status

approved