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

%I A192210 #10 Oct 14 2012 09:08:09
%S A192210 1,4,10,26,80,194,504,1442,3710,9536,26842,69014,178704,496602,
%T A192210 1316204,3377206,9242898,24629944,63304540,172497622,462822414,
%U A192210 1210912388,3177522724,8736822276,22617998204,59776061150,163702751968,433787373560
%N A192210 a(n) = sum of unsigned coefficients in (1+x+x^2-x^3)^n.
%C A192210 What is the behavior of this sequence? Does there exist a g.f.?
%C A192210 It would be nice to know the (more accurate) values of the following limits:
%C A192210 (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...
%C A192210 (2) Limit a(n)^(1/n) seems to exist near 2.6637...
%C A192210 (3) Limit a(n+1)/a(n) does not seem to be unique; attractors seem to exist near 2.66...
%H A192210 Vincenzo Librandi, <a href="/A192210/b192210.txt">Table of n, a(n) for n = 0..300</a>
%e A192210 Illustrate the coefficients in (1+x+x^2-x^3)^n by:
%e A192210 n=0: [1];
%e A192210 n=1: [1, 1, 1, -1];
%e A192210 n=2: [1, 2, 3, 0, -1, -2, 1];
%e A192210 n=3: [1, 3, 6, 4, 0, -6, -2, 0, 3, -1];
%e A192210 n=4: [1, 4, 10, 12, 7, -8, -12, -8, 7, 4, 2, -4, 1];
%e A192210 n=5: [1, 5, 15, 25, 25, 1, -25, -35, -5, 15, 21, -5, -5, -5, 5, -1];
%e A192210 n=6: [1, 6, 21, 44, 60, 36, -24, -84, -66, 0, 66, 36, -4, -36, 0, 4, 9, -6, 1];
%e A192210 n=7: [1, 7, 28, 70, 119, 119, 28, -132, -210, -126, 84, 168, 98, -70, -76, -28, 49, 7, 0, -14, 7, -1]; ...
%e A192210 This sequence gives the sums of the absolute values of the coefficients for n>=0.
%t A192210 Table[Total[Abs[CoefficientList[Expand[(1+x+x^2-x^3)^n],x]]],{n,0,30}] (* _Harvey P. Dale_, Oct 12 2012 *)
%o A192210 (PARI) {a(n)=sum(k=0,3*n,abs(polcoeff((1+x+x^2-x^3)^n,k)))}
%Y A192210 Cf. A192205, A084611, A084613, A084615, A084775, A084776, A084777, A084778, A084779, A084780.
%K A192210 nonn
%O A192210 0,2
%A A192210 _Paul D. Hanna_, Jun 25 2011