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.
Original entry on oeis.org
1, -2, 1, 576, 46225, 2000989041197056, 198828783273803025550632280753863681, 6104549033356152351183622743336946156997116945571290671544232012635281247174656
Offset: 0
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.
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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);
-
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 *)
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# See LINKS
A247209
Number of terms in generalized Swinnerton-Dyer polynomials.
Original entry on oeis.org
1, 2, 6, 35, 495, 20349, 2760681, 1329890705, 2353351951665, 15481400876017505, 379554034822178909121, 34676179189150610052785025, 11806724418359403847522843860225, 14998128029851443976142151169687652865, 71221988684076361563783957084457295633613825
Offset: 0
a(3) = 35. For the three numbers a, b, c, the general Swinnerton-Dyer polynomial is
(sqrt(a)+sqrt(b)+sqrt(c)-z)(-sqrt(a)+sqrt(b)+sqrt(c)-z)(sqrt(a)-sqrt(b)+sqrt(c)-z)(-sqrt(a)-sqrt(b)+sqrt(c)-z)(sqrt(a)+sqrt(b)-sqrt(c)-z)(-sqrt(a)+sqrt(b)-sqrt(c)-z)(sqrt(a)-sqrt(b)-sqrt(c)-z)(-sqrt(a)-sqrt(b)-sqrt(c)-z)
which expands to
a^4-4a^3b+6a^2b^2-4ab^3+b^4-4a^3c+4a^2bc+4ab^2c-4b^3c+6a^2c^2+4abc^2+6b^2c^2-4ac^3-4bc^3+c^4- 4a^3z^2+4a^2bz^2+4ab^2z^2-4b^3z^2+4a^2cz^2-40abcz^2+ 4b^2cz^2+4ac^2z^2+4bc^2z^2-4c^3z^2+6a^2z^4+4abz^4+ 6b^2z^4+4acz^4+4bcz^4+6c^2z^4-4az^6-4bz^6-4cz^6+z^8
with 35 terms.
- Alois P. Heinz, Table of n, a(n) for n = 0..60
- Allan Berele and Stefan Catoiu, Rationalizing Denominators, Mathematics Magazine, Vol. 88, No. 2 (2015), pp. 121-136.
- Les Tablettes du Chercheur, Problem 21, Solution to problem 21, Addition to problem 21, pp. 4, 30 and 64, 1892 (in French).
- Eric Weisstein's World of Mathematics, Swinnerton-Dyer Polynomial.
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a:= n-> (t-> binomial(t+n, t))(2^(n-1)):
seq(a(n), n=0..14); # Alois P. Heinz, Nov 28 2024
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a[n_]:= Module[{a,x}, Length@Fold[Expand[(#1 /. x -> x + #2) (#1 /. x -> x - #2)] &, x, Sqrt[a /@ Range[n]]]]; a[0] = 1; Array[a, 5, 0] (* or *)
a[n_]:= Binomial[2^(n - 1) + n, 2^(n - 1)]; Array[a, 10, 0]
A255587
Maximum absolute value of the coefficients of the n-th Swinnerton-Dyer polynomial.
Original entry on oeis.org
1, 2, 10, 960, 13950764, 255690851718529024, 1771080720430629161685158978892152599456, 857834471403601877816627441633642526746656338035964968069692458744011458425706833248256
Offset: 0
SwinnertonDyerP(3, x) = x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, hence a(3) = 960.
- Roman E. Maeder, Programming in Mathematica, Addison-Wesley, 1990, page 105.
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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
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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}]
A376552
Square root of the product of all sums and differences of the square roots of the first n primes.
Original entry on oeis.org
1, 24, 215, 44732416, 445902212680990209, 2470738560300573839567485058051752329216, 194775879942444285383551347529278187374780378665463617801353369255538909241232419740031
Offset: 2
The Swinnerton-Dyer polynomial for n=1 is x^2 - 2, which has negative constant term, so we skip n = 1.
For n = 2, the Swinnerton-Dyer polynomial is (x + sqrt(2) + sqrt(3)) * (x + sqrt(2) - sqrt(3)) * (x - sqrt(2) + sqrt(3)) * (x - sqrt(2) - sqrt(3)) = x^4 - 10*x^2 + 1, so a(2) = 1.
For n = 3, the Swinnerton-Dyer polynomial is x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, so a(3) = 24.
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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-> isqrt(coeff(p(n), x, 0)):
seq(a(n), n=2..8); # Alois P. Heinz, Nov 28 2024
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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]] // Sqrt // Floor;
Table[a[n], {n, 2, 10}] (* Jean-François Alcover, Jul 02 2025, after Alois P. Heinz *)
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# See LINKS.
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