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

-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

1,1

References

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

Links

Table of n, a(n) for n=1..26.

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

Example

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

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 *)

Prog

(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<x> := 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

Crossrefs

Cf. A247209, A354913 (left column).

Sequence in context: A348455 A348453 A345748 * A262226 A298158 A154989

Adjacent sequences: A153728 A153729 A153730 * A153732 A153733 A153734

Keyword

sign,tabf

Author

Eric W. Weisstein, Dec 31 2008

Status

approved