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

%I A354913 #84 Feb 16 2025 08:34:03
%S A354913 1,-2,1,576,46225,2000989041197056,
%T A354913 198828783273803025550632280753863681,
%U A354913 6104549033356152351183622743336946156997116945571290671544232012635281247174656
%N A354913 Nonzero coefficient with smallest index in polynomial p_n(x), where p_n(x) = Product_{i={-1,1}} p_{n-1}(x + i*sqrt(prime(n))), starting with p_0(x) = x.
%C A354913 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.
%C A354913 For n>=1, a(n) is also the product of all 2^n sums and differences of the square roots of the first n primes.
%C A354913 For n>=2, each term is a perfect square; the square roots are 1, 24, 215, 44732416, ... (A376552).
%H A354913 Alois P. Heinz, <a href="/A354913/b354913.txt">Table of n, a(n) for n = 0..10</a>.
%H A354913 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A354913.py">Python program</a>.
%H A354913 Alois P. Heinz, <a href="/A354913/a354913_1.jpg">Scaled plot of polynomials p(0) ... p(4)</a>.
%H A354913 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>.
%H A354913 Wikipedia, <a href="https://en.wikipedia.org/wiki/Swinnerton-Dyer_polynomial">Swinnerton-Dyer polynomial</a>.
%F A354913 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))).
%e A354913 The first polynomials p_0(x) ... p_3(x) are:
%e A354913   x,
%e A354913   x^2 -2,
%e A354913   x^4 -10*x^2 +1,
%e A354913   x^8 -40*x^6 +352*x^4 -960*x^2 +576,
%e A354913   so the sequence starts 1, -2, 1, 576.
%e A354913 p_0(x) has 2^0 = 1 root: 0 (empty sum).
%e A354913 p_1(x) has 2^1 = 2 roots: -sqrt(2), sqrt(2); their product gives a(1) = -2.
%e A354913 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.
%p A354913 p:= proc(n) option remember; expand(`if`(n=0, x, mul(
%p A354913       subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
%p A354913     end:
%p A354913 a:= n-> coeff(p(n), x, 1-signum(n)):
%p A354913 seq(a(n), n=0..8);
%t A354913 p[n_] := p[n] = Expand[If[n == 0, x, Product[
%t A354913      p[n-1] /. x -> x+i*Sqrt[Prime[n]], {i, {1, -1}}]]];
%t A354913 a[n_] := Coefficient[p[n], x, 1-Sign[n]];
%t A354913 Table[a[n], {n, 0, 8}] (* _Jean-François Alcover_, Jun 25 2022, after _Alois P. Heinz_ *)
%o A354913 (Python) # See LINKS
%Y A354913 Left column of A153731.
%Y A354913 Cf. A000007, A000040, A000079, A000290, A255587, A262396, A376552.
%K A354913 sign
%O A354913 0,2
%A A354913 _Alois P. Heinz_, Jun 12 2022