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Let $ϵ = (ϵ_{n})_{n \geq 0}$ be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations: $ϵ_{0} = 1, ϵ_{2 n} = ϵ_{n}, ϵ_{2 n + 1} = 1 - ϵ_{n} .$ We consider ${| ℰ_{n}^{p} |}_{n \geq 1, p \geq 0}$ , the double sequence of Hankel determinants (modulo 2) associated with the Thue-Morse sequence. Together with three other sequences, it obeys a set of sixteen recurrence equations. It is shown to be automatic. Applications are given, namely to combinatorial properties of the Thue-Morse sequence and to the existence of certain Padé approximants of the power series $\sum_{n \geq 0} (- 1)^{ϵ_{n}} x^{n}$ .

Hankel matrices and lattice paths.

Woan, Wen-Jin (2001)

Journal of Integer Sequences [electronic only]

Hankel transforms of linear combinations of Catalan numbers.

Dougherty, Michael, French, Christopher, Saderholm, Benjamin, Qian, Wenyang (2011)

Journal of Integer Sequences [electronic only]

Harmonic number identities via Euler's transform.

Boyadzhiev, Khristo N. (2009)

Journal of Integer Sequences [electronic only]

Hecke operators and Lambert series.

L. Alayne Parson, Kenneth H. Rosen (1981)

Mathematica Scandinavica

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_{m}$ be the mth coefficient of the square f(x)² of a unimodular...