A195417 a(n) = sum of the absolute values of the coefficients in (1+x-x^3)^n.

1, 3, 9, 25, 63, 171, 471, 1193, 3209, 8841, 22537, 60393, 165489, 424451, 1134849, 3093893, 7987289, 21288933, 57973543, 150409931, 399788279, 1085870421, 2831439537, 7501994181, 20330726077, 53274740933, 140615421393, 380366331549, 1002078031627, 2634444052559

Offset: 0

Author

Paul D. Hanna, Sep 17 2011

Keywords

• nonn

Comments

What is the limit a(n)^(1/n)? The limit appears to be near 2.657...

The limit a(n+1)/a(n) does not exist, but seems to cycle between attractors.

Examples

			Coefficients in (1+x-x^3)^n begin:
n=0: [1];
n=1: [1,1,0,-1];
n=2: [1,2,1,-2,-2,0,1];
n=3: [1,3,3,-2,-6,-3,3,3,0,-1];
n=4: [1,4,6,0,-11,-12,2,12,6,-4,-4,0,1];
n=5: [1,5,10,5,-15,-29,-10,25,30,0,-20,-10,5,5,0,-1];
n=6: [1,6,15,14,-15,-54,-44,30,84,40,-45,-60,-5,30,15,-6,-6,0,1];
n=7: [1,7,21,28,-7,-84,-112,1,168,168,-35,-189,-105,70,105,14,-42,-21,7,7,0,-1];
n=8: [1,8,28,48,14,-112,-224,-104,253,448,132,-392,-462,0,364,224,-98,-168,-28,56,28,-8,-8,0,1]; ...
where the sum of the absolute values of the coefficients generate this sequence.
The limit a(n)^(1/n) seems to exist:
a(1000)^(1/1000) = 2.65862772...
a(2000)^(1/2000) = 2.65827933...
a(3000)^(1/3000) = 2.65816317...
a(4000)^(1/4000) = 2.65810624...
a(5000)^(1/5000) = 2.65807183...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A084611.

Programs

Mathematica

Table[Total[Abs[CoefficientList[Expand[(1+x-x^3)^n],x]]],{n,0,30}] (* Harvey P. Dale, Nov 26 2011 *)

PARI

{a(n)=local(V=Vec((1+x-x^3+O(x^(3*n+1)))^n));abs(V)*vector(#V,i,1)~}