A192205 a(n) = sum of absolute values of coefficients in (1-x-x^2+x^3)^n.

1, 4, 12, 36, 116, 344, 1104, 3280, 10456, 31152, 98804, 295988, 935876, 2811540, 8870324, 26695724, 84060148, 253376840, 796635360, 2404558304, 7549431884, 22820942416, 71541295984, 216562743948, 677938097756, 2054922521644

Offset

0,2

Comments

Conjecture: limit a(n)^(1/n) = 16*sqrt(3)/9 = 3.079201..., which is substantiated by the observation that the sums of the coefficients squared in (1-x-x^2+x^3)^n equals binomial(4n,n) (cf. A005810).

Links

Table of n, a(n) for n=0..25.

Example

The triangle A227964 of coefficients in (1+x-x^2-x^3)^n begins:
n=0: [1];
n=1: [1, -1, -1, 1];
n=2: [1, -2, -1, 4, -1, -2, 1];
n=3: [1, -3, 0, 8, -6, -6, 8, 0, -3, 1];
n=4: [1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1];
n=5: [1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1];
n=6: [1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 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, Mar 03 2013 *)

Prog

(PARI) {a(n)=sum(k=0, 3*n, abs(polcoeff((1-x-x^2+x^3)^n, k)))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A192210, A084611, A084613, A084615, A084775, A084776, A084777, A084778, A084779, A084780.

Cf. A227964 (triangle).

Sequence in context: A290899 A290905 A000781 * A055395 A113990 A231179

Adjacent sequences: A192202 A192203 A192204 * A192206 A192207 A192208

Keyword

nonn

Author

Paul D. Hanna, Jun 25 2011

Status

approved