A255587 - OEIS

%I #14 Jun 18 2022 12:11:20

%S 1,2,10,960,13950764,255690851718529024,

%T 1771080720430629161685158978892152599456,

%U 857834471403601877816627441633642526746656338035964968069692458744011458425706833248256

%N Maximum absolute value of the coefficients of the n-th Swinnerton-Dyer polynomial.

%D Roman E. Maeder, Programming in Mathematica, Addison-Wesley, 1990, page 105.

%H Alois P. Heinz, <a href="/A255587/b255587.txt">Table of n, a(n) for n = 0..10</a>

%H MathOverflow, <a href="http://mathoverflow.net/questions/106907">What bounds can we establish on coefficients of Swinnerton-Dyer polynomials?</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>

%e SwinnertonDyerP(3, x) = x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, hence a(3) = 960.

%p p:= proc(n) option remember; expand(`if`(n=0, x, mul(

%p       subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))

%p     end:

%p a:= n-> max(map(abs, [coeffs(p(n))])):

%p seq(a(n), n=0..8);  # _Alois P. Heinz_, Jun 12 2022

%t SwinnertonDyerP[0, x_] := x; SwinnertonDyerP[n_, x_] := Module[{sd, srp = Sqrt[Prime[n]] }, sd[y_] = SwinnertonDyerP[n - 1, y]; Expand[sd[x + srp] sd[x - srp]]]; a[n_] := CoefficientList[SwinnertonDyerP[n, x], x^2] // Abs // Max; Table[a[n], {n, 1, 10}]

%Y Cf. A153731, A354913.

%K nonn

%O 0,2

%A _Jean-François Alcover_, Feb 27 2015

%E a(0)=1 prepended by _Alois P. Heinz_, Jun 12 2022