A166445 Hankel transform of A025276.

1, 0, -1, 1, 3, 0, -3, 1, 5, 0, -5, 1, 7, 0, -7, 1, 9, 0, -9, 1, 11, 0, -11, 1, 13, 0, -13, 1, 15, 0, -15, 1, 17, 0, -17, 1, 19, 0, -19, 1, 21, 0, -21, 1, 23, 0, -23, 1, 25, 0, -25, 1, 27, 0, -27, 1, 29, 0, -29, 1, 31, 0, -31, 1, 33, 0, -33, 1, 35, 0, -35, 1

Offset

0,5

Links

Felix Fröhlich, Table of n, a(n) for n = 0..10000

Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.

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

Formula

G.f.: (1-x+x^2+x^4)/((1-x)*(1+x^2)^2).

a(n) = (1/2)*(1 + cos((n+1)*Pi/2) + (n+1)*sin((n+1)*Pi/2)). - Harvey P. Dale, Nov 21 2014; corrected by Bernard Schott, Jun 27 2022

For n >= 0: a(4n) = 2n+1; a(4n+1) = 0; a(4n+2) = -a(4n) = -2n-1; a(4n+3) = 1. - Bernard Schott, Jun 27 2022

a(n) - a(n-1) = A127365(n+1). - R. J. Mathar, Jul 01 2024

E.g.f.: (exp(x) + cos(x) - (1 + x)*sin(x))/2. - Stefano Spezia, Jul 14 2024

a(n) = (1/2)*(1 - A056594(n) - A056594(n-1) + 2*(-1)^floor(n/2) * A027656(n)). - G. C. Greubel, Jul 27 2024

Mathematica

LinearRecurrence[{1, -2, 2, -1, 1}, {1, 0, -1, 1, 3}, 80] (* Harvey P. Dale, Nov 21 2014 *)

Prog

(PARI) Vec((1-x+x^2+x^4)/((1-x)*(1+x^2)^2) + O(x^80)) \\ Felix Fröhlich, Jun 28 2022
(Magma)
R<x>:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (1-x+x^2+x^4)/((1-x)*(1+x^2)^2) )); // G. C. Greubel, Jul 27 2024
(SageMath)
def A166445_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2+x^4)/((1-x)*(1+x^2)^2) ).list()
A166445_list(80) # G. C. Greubel, Jul 27 2024

Crossrefs

Cf. A025276, A027656, A056594, A127365.

Sequence in context: A376627 A237595 A322575 * A374139 A298082 A085919

Adjacent sequences: A166442 A166443 A166444 * A166446 A166447 A166448

Keyword

easy,sign

Author

Paul Barry, Oct 13 2009

Status

approved