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A255587 Maximum absolute value of the coefficients of the n-th Swinnerton-Dyer polynomial.

%I A255587 #15 Feb 16 2025 08:33:25
%S A255587 1,2,10,960,13950764,255690851718529024,
%T A255587 1771080720430629161685158978892152599456,
%U A255587 857834471403601877816627441633642526746656338035964968069692458744011458425706833248256
%N A255587 Maximum absolute value of the coefficients of the n-th Swinnerton-Dyer polynomial.
%D A255587 Roman E. Maeder, Programming in Mathematica, Addison-Wesley, 1990, page 105.
%H A255587 Alois P. Heinz, <a href="/A255587/b255587.txt">Table of n, a(n) for n = 0..10</a>
%H A255587 MathOverflow, <a href="http://mathoverflow.net/questions/106907">What bounds can we establish on coefficients of Swinnerton-Dyer polynomials?</a>
%H A255587 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>
%e A255587 SwinnertonDyerP(3, x) = x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, hence a(3) = 960.
%p A255587 p:= proc(n) option remember; expand(`if`(n=0, x, mul(
%p A255587       subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
%p A255587     end:
%p A255587 a:= n-> max(map(abs, [coeffs(p(n))])):
%p A255587 seq(a(n), n=0..8);  # _Alois P. Heinz_, Jun 12 2022
%t A255587 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 A255587 Cf. A153731, A354913.
%K A255587 nonn
%O A255587 0,2
%A A255587 _Jean-François Alcover_, Feb 27 2015
%E A255587 a(0)=1 prepended by _Alois P. Heinz_, Jun 12 2022