A192205 -id:A192205 - cp's OEIS frontend

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

Paul D. Hanna, Jun 25 2011

nonn

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...

			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.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

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

Table[Total[Abs[CoefficientList[Expand[(1+x+x^2-x^3)^n],x]]],{n,0,30}] (* Harvey P. Dale, Oct 12 2012 *)

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

A227964 Triangle where the g.f. of row n equals (1-x-x^2+x^3)^n and terms T(n,k) are read by rows n>=0, k=0..3*n.

Original entry on oeis.org

1, 1, -1, -1, 1, 1, -2, -1, 4, -1, -2, 1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1, 1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1, 1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1, 1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1

Offset: 0

Paul D. Hanna, Aug 01 2013

			Triangle begins:
1;
1, -1, -1, 1;
1, -2, -1, 4, -1, -2, 1;
1, -3, 0, 8, -6, -6, 8, 0, -3, 1;
1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1;
1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1;
1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1;
1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..2500

Cf. A192205.

{T(n,k)=polcoeff((1-x-x^2+x^3 +x*O(x^k))^n,k)}
for(n=0,10,for(k=0,3*n,print1(T(n,k),", "));print(""))

Sum_{k=0..3*n} |T(n,k)| = A192205(n).
Sum_{k=0..3*n} T(n,k)^2 = binomial(4*n,n).
Sum_{k=0..3*n} T(n,k) * binomial(3*n,k) = (-1)^n * binomial(4*n,n).
Sum_{k=0..3*n} T(n,k) * binomial(2*n+k,k) = 2^n.
Sum_{k=0..3*n} T(n,k) * binomial(3*n+k,k) = A008288(3*n,n), where A008288 is the Delannoy array (see A026001).

cp's OEIS Frontend

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

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