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

1, 2, 10, 960, 13950764, 255690851718529024, 1771080720430629161685158978892152599456, 857834471403601877816627441633642526746656338035964968069692458744011458425706833248256

Offset

0,2

References

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10

MathOverflow, What bounds can we establish on coefficients of Swinnerton-Dyer polynomials?

Eric Weisstein's World of Mathematics, Swinnerton-Dyer Polynomial

Example

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

Maple

p:= proc(n) option remember; expand(`if`(n=0, x, mul(
      subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
    end:
a:= n-> max(map(abs, [coeffs(p(n))])):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 12 2022

Mathematica

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}]

Crossrefs

Cf. A153731, A354913.

Sequence in context: A222420 A263921 A290495 * A137890 A074333 A008559

Adjacent sequences: A255584 A255585 A255586 * A255588 A255589 A255590

Keyword

nonn

Author

Jean-François Alcover, Feb 27 2015

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 12 2022

Status

approved