A359738 a(n) = [x^n] (2*x^4 + 2*x^3 + 2*x^2 + x + 1)/(x^2 + 1).

1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1

Offset

0

Links

Table of n, a(n) for n=0..74.

Madeline Beals-Reid, A Quadratic Relation in the Bernoulli Numbers, The Pump Journal of Undergraduate Research, 6 (2023), 29-39.

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

Formula

Let B(x) = x/(1 - exp(-x)), the e.g.f. of the Bernoulli numbers with B(1) = 1/2.

a(n) = signum([x^n] B(x)^2) = signum([x^n] z^2 / (exp(-z) - 1)^2).

a(n) = signum([x^n] (x + 1)*B(x) - x*B'(x)).

a(n) = A057077(n-3), n>2. - R. J. Mathar, Jan 27 2025

E.g.f.: x*(2 + x) + cos(x) - sin(x). - Stefano Spezia, Jan 27 2025

Maple

ogf := (2*z^4 + 2*z^3 + 2*z^2 + z + 1)/(z^2 + 1):
ser := series(ogf, z, 100): seq(coeff(ser, z, n), n = 0..74);

Crossrefs

Cf. A057077, A087960, A266591, A100615.

Sequence in context: A319116 A337004 A343785 * A360710 A269529 A325931

Adjacent sequences: A359735 A359736 A359737 * A359739 A359740 A359741

Keyword

sign,easy

Author

Peter Luschny, Jan 23 2023

Status

approved