A128213 Expansion of (1-x+2x^2-2x^3)/(1-x+x^2)^2.

1, 1, 1, -1, -4, -4, 1, 7, 7, -1, -10, -10, 1, 13, 13, -1, -16, -16, 1, 19, 19, -1, -22, -22, 1, 25, 25, -1, -28, -28, 1, 31, 31, -1, -34, -34, 1, 37, 37, -1, -40, -40, 1, 43, 43, -1, -46, -46, 1, 49, 49, -1, -52, -52, 1, 55, 55, -1, -58, -58, 1, 61, 61, -1

Offset: 0

Author

Paul Barry, Feb 19 2007

Keywords

• easy
• sign

Comments

a(n+1) is the Hankel transform of {1,0,1,3,9,28,90,297,1001,3432,11934,...}, cf. A000245.

Binomial transform of A128214.

a(n+2) is the Hankel transform of A014138. - Paul Barry, Mar 15 2008

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Programs

Mathematica

Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == y[m] - y[m - 1], y[0] == 1, y[1] == n}]][n], {n, 0, 100}] (* Benedict W. J. Irwin, Nov 05 2016 *)

PARI

Vec((1-x+2*x^2-2*x^3)/(1-x+x^2)^2 + O(x^100)) \\ Michel Marcus, May 31 2014

Formula

a(n) = cos(Pi*n/3) + (2n/sqrt(3)-1/sqrt(3))*sin(Pi*n/3).

a(n) = y(n,n), where y(m+1,n) = y(m,n) - y(m-1,n), with y(0,n)=1 and y(1,n)=n. - Benedict W. J. Irwin, Nov 05 2016