A127630 Expansion of (1+x-x^2-x^3)/(1+x^2)^2.

1, 1, -3, -3, 5, 5, -7, -7, 9, 9, -11, -11, 13, 13, -15, -15, 17, 17, -19, -19, 21, 21, -23, -23, 25, 25, -27, -27, 29, 29, -31, -31, 33, 33, -35, -35, 37, 37, -39, -39, 41, 41, -43, -43, 45, 45, -47, -47, 49, 49, -51, -51, 53, 53, -55, -55

Offset: 0

Author

Paul Barry, Jan 20 2007

Keywords

• sign
• easy

Comments

Hankel transform of A045621(n+1).

Links

• Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Crossrefs

Cf. A109613.

Programs

Mathematica

CoefficientList[Series[(1+x-x^2-x^3)/(1+x^2)^2,{x,0,100}],x] (* or *) LinearRecurrence[{0,-2,0,-1},{1,1,-3,-3},100] (* Harvey P. Dale, Nov 18 2020 *)

Formula

a(n) = (-1)^binomial(n,2) * ( 2*floor(n/2)+1 ).

a(n) = (n + 1 - (n mod 2))*(-1)^floor(n/2). [Wesley Ivan Hurt, Jun 30 2013]

G.f.: 1/(G(0) - x), where G(k) = x*(2*k+1) - (2*k-1)/( 1 + x/( 1 - x*(2*k+3)/G(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Dec 27 2013