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, 57, 57, -59, -59, 61, 61

Offset

0,3

Comments

Hankel transform of A045621(n+1).

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

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

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

E.g.f.: (1+x)*cos(x) - x*sin(x). - Amiram Eldar, Dec 17 2025

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 *)

Crossrefs

Cf. A045621, A109613.

Sequence in context: A340836 A293701 A296063 * A267458 A109613 A063196

Adjacent sequences: A127627 A127628 A127629 * A127631 A127632 A127633

Keyword

sign,easy

Author

Paul Barry, Jan 20 2007

Status

approved