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A128213 Expansion of (1-x+2x^2-2x^3)/(1-x+x^2)^2. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

Table of n, a(n) for n=0..63.

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

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

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 *)

PROG

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

CROSSREFS

Sequence in context: A337191 A341863 A047213 * A171716 A211788 A318732

Adjacent sequences:  A128210 A128211 A128212 * A128214 A128215 A128216

KEYWORD

easy,sign

AUTHOR

Paul Barry, Feb 19 2007

STATUS

approved

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Last modified May 17 22:15 EDT 2021. Contains 343992 sequences. (Running on oeis4.)