A153731 Triangle read by rows: nonzero coefficients of Swinnerton-Dyer polynomials.

1, -2, 1, 1, -10, 1, 576, -960, 352, -40, 1, 46225, -5596840, 13950764, -7453176, 1513334, -141912, 6476, -136, 1, 2000989041197056, -44660812492570624, 183876928237731840, -255690851718529024, 172580952324702208, -65892492886671360, 15459151516270592

Offset: 0

Author

Eric W. Weisstein, Dec 31 2008

Keywords

• sign
• look
• tabf

Comments

Within each row, coefficients are listed in order of increasing degree. The n-th row lists the coefficients of the polynomial corresponding to the set {2, 3, ..., prime(n)}. All odd-degree terms have coefficient 0.

Examples

			The first few rows are:
[0]      1;
[1]     -2,        1;
[2]      1,      -10,        1;
[3]    576,     -960,      352,      -40,       1;
[4]  46225, -5596840, 13950764, -7453176, 1513334, -141912, 6476, -136, 1;
  ....
  x, -2 + x^2, 1 - 10*x^2 + x^4, 576 - 960*x^2 + 352*x^4 - 40*x^6 + x^8, ...
		

References

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

Links

• Alois P. Heinz, Rows n = 0..10, flattened
• Lucas A. Brown, Python program.
• Eric Weisstein's World of Mathematics, Swinnerton-Dyer Polynomial.
• Wikipedia, Swinnerton-Dyer polynomial.

Crossrefs

Cf. A247209, A354913 (left column).

Programs

Julia

using Nemo
function A153731Row(n)
    R, x = PolynomialRing(ZZ, "x")
    p = swinnerton_dyer(n, x)
    [coeff(p, j) for j in 0:2:2^n] end
for n in 1:4 A153731Row(n) |> println end # Peter Luschny, Mar 13 2018

Magma

// Note that Magma, like Mathworld, defines the polynomials for n >= 1.
P := PolynomialRing(IntegerRing());
for n := 1 to 5 do
    p := SwinnertonDyerPolynomial(n);
    [c : c in Coefficients(p) | not IsZero(c)];
end for;  // Peter Luschny, Jun 12 2022

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:
T:= n-> ListTools[Reverse]([coeffs(p(n))])[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Nov 28 2024

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] ] ]; row[n_] := CoefficientList[ SwinnertonDyerP[n, x], x^2]; Table[row[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Nov 09 2012 *)
(* Second program: *)
SwinnertonDyerP[n_Integer?Positive, x_] :=
    Block[{arg, poly, i},
        args = Outer[Times, Table[Sqrt[Prime[i]], {i, n}], {-1, 1}];
        poly = Outer[Plus, {x}, Sequence @@ args];
        Expand[Times @@ Flatten[poly]]]
Table[Select[CoefficientList[SwinnertonDyerP[n, x], x], # != 0 &], {n, 1, 4}] // TableForm  (* Peter Luschny, Jun 12 2022 *)

Python

# See LINKS

Extensions

One term (row 0) prepended by Alois P. Heinz, Nov 28 2024