A331691 Resultant of the Shapiro polynomials P_n(x) and Q_n(x).

1, 2, -16, 2048, -67108864, 144115188075855872, -1329227995784915872903807060280344576, 226156424291633194186662080095093570025917938800079226639565593765455331328

Offset: 0

Author

Kevin Ryde, Jan 24 2020

Keywords

• sign
• easy

Comments

The Shapiro polynomials P_n(x) and Q_n(x) are defined by P_0(x) = Q_0(x) = 1 and then mutual recurrences P_{n+1}(x) = P_n(x) + x^(2^n)*Q_n(x) and Q_{n+1}(x) = P_n(x) - x^(2^n)*Q_n(x). The coefficients of P are the Golay-Rudin-Shapiro sequence A020985. a(n) is the polynomial resultant R(P_n(x),Q_n(x)) as considered by Brillhart and Carlitz.

Links

• John Brillhart and L. Carlitz, Note on the Shapiro Polynomials, Proceedings of the American Mathematical Society, volume 25, number 1, May 1970, pages 114-118. Also at JSTOR. See A001782 for a scanned copy.
• Harold S. Shapiro, Extremal Problems for Polynomials and Power Series, Masters Thesis, Massachusetts Institute of Technology, 1951. See pages 40-41.

Crossrefs

Cf. A016031 (absolute values), A001782 (discriminant).

Programs

PARI

a(n) = if(n==0,1, -(-2)^(2^(n+1) - n - 2));

PARI

a(n) = my(P=1,Q=1); for(i=0,n-1, [P,Q]=[P+x^(2^i)*Q, P-x^(2^i)*Q]); polresultant(P,Q);

Formula

a(n) = (-1)^(n-1) * 2^(2^(n+1) - n - 2) for n >= 1 [Brillhart and Carlitz theorem 2].

a(n) = (-1)^(n-1) * A016031(n+2) for n >= 1.

a(n) = - 2^(2^n-1) * a(n-1) for n >= 2 [Brillhart and Carlitz in proof of theorem 2].