A354913 Nonzero coefficient with smallest index of polynomial p_n(x), where p_n(x) = Product_{i={-1,1}} p_{n-1}(x + i*sqrt(prime(n))), starting p_0(x) = x.

1, -2, 1, 576, 46225, 2000989041197056, 198828783273803025550632280753863681, 6104549033356152351183622743336946156997116945571290671544232012635281247174656

Offset

0,2

Comments

These polynomials are known as Swinnerton-Dyer polynomials: p_n(x) is of degree 2^n and has 2^n = A000079(n) real roots, which are the sums and differences of the square roots of the first n primes.

For n>=1 also product of the sums and differences of the square roots of the first n primes, combined in all possible ways.

For n>=2 each term a(n) is a perfect square, the square roots are 1, 24, 215, 44732416, ... .

Links

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

Alois P. Heinz, Scaled plot of polynomials p(0) ... p(4)

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

Wikipedia, Swinnerton-Dyer polynomials

Formula

a(n) = [x^A000007(n)] p_n(x), with p_n(x) = Product_{v={-1,1}^n} (x + Sum_{i=1..n} v[i]*sqrt(prime(i))).

Example

The first polynomials p_0(x) ... p_3(x) are:
  x,
  x^2 -2,
  x^4 -10*x^2 +1,
  x^8 -40*x^6 +352*x^4 -960*x^2 +576,
  so the sequence starts 1, -2, 1, 576.
p_0(x) has 2^0 = 1 root: 0 (empty sum).
p_1(x) has 2^1 = 2 roots: -sqrt(2), sqrt(2); their product gives a(1) = -2.
p_2(x) has 2^2 = 4 roots: -sqrt(2)-sqrt(3), -sqrt(2)+sqrt(3), sqrt(2)-sqrt(3), sqrt(2)+sqrt(3); their product gives a(2) = 1.

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-> coeff(p(n), x, 1-signum(n)):
seq(a(n), n=0..8);

Mathematica

p[n_] := p[n] = Expand[If[n == 0, x, Product[
     p[n-1] /. x -> x+i*Sqrt[Prime[n]], {i, {1, -1}}]]];
a[n_] := Coefficient[p[n], x, 1-Sign[n]];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jun 25 2022, after Alois P. Heinz *)

Crossrefs

Left column of A153731.

Cf. A000007, A000040, A000079, A000290, A255587, A262396.

Sequence in context: A178393 A272538 A087037 * A036109 A240234 A367996

Adjacent sequences: A354910 A354911 A354912 * A354914 A354915 A354916

Keyword

sign

Author

Alois P. Heinz, Jun 12 2022

Status

approved