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

%I A153731 #49 Feb 16 2025 08:33:09
%S A153731 1,-2,1,1,-10,1,576,-960,352,-40,1,46225,-5596840,13950764,-7453176,
%T A153731 1513334,-141912,6476,-136,1,2000989041197056,-44660812492570624,
%U A153731 183876928237731840,-255690851718529024,172580952324702208,-65892492886671360,15459151516270592
%N A153731 Triangle read by rows: nonzero coefficients of Swinnerton-Dyer polynomials.
%C A153731 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.
%D A153731 Roman E. Maeder. Programming in Mathematica, Addison-Wesley, 1990, page 105.
%H A153731 Alois P. Heinz, <a href="/A153731/b153731.txt">Rows n = 0..10, flattened</a>
%H A153731 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A153731.py">Python program</a>.
%H A153731 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>.
%H A153731 Wikipedia, <a href="https://en.wikipedia.org/wiki/Swinnerton-Dyer_polynomial">Swinnerton-Dyer polynomial</a>.
%e A153731 The first few rows are:
%e A153731 [0]      1;
%e A153731 [1]     -2,        1;
%e A153731 [2]      1,      -10,        1;
%e A153731 [3]    576,     -960,      352,      -40,       1;
%e A153731 [4]  46225, -5596840, 13950764, -7453176, 1513334, -141912, 6476, -136, 1;
%e A153731   ....
%e A153731   x, -2 + x^2, 1 - 10*x^2 + x^4, 576 - 960*x^2 + 352*x^4 - 40*x^6 + x^8, ...
%p A153731 p:= proc(n) option remember; expand(`if`(n=0, x, mul(
%p A153731       subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
%p A153731     end:
%p A153731 T:= n-> ListTools[Reverse]([coeffs(p(n))])[]:
%p A153731 seq(T(n), n=0..5);  # _Alois P. Heinz_, Nov 28 2024
%t A153731 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 *)
%t A153731 (* Second program: *)
%t A153731 SwinnertonDyerP[n_Integer?Positive, x_] :=
%t A153731     Block[{arg, poly, i},
%t A153731         args = Outer[Times, Table[Sqrt[Prime[i]], {i, n}], {-1, 1}];
%t A153731         poly = Outer[Plus, {x}, Sequence @@ args];
%t A153731         Expand[Times @@ Flatten[poly]]]
%t A153731 Table[Select[CoefficientList[SwinnertonDyerP[n, x], x], # != 0 &], {n, 1, 4}] // TableForm  (* _Peter Luschny_, Jun 12 2022 *)
%o A153731 (Julia)
%o A153731 using Nemo
%o A153731 function A153731Row(n)
%o A153731     R, x = PolynomialRing(ZZ, "x")
%o A153731     p = swinnerton_dyer(n, x)
%o A153731     [coeff(p, j) for j in 0:2:2^n] end
%o A153731 for n in 1:4 A153731Row(n) |> println end # _Peter Luschny_, Mar 13 2018
%o A153731 (Magma) // Note that Magma, like Mathworld, defines the polynomials for n >= 1.
%o A153731 P<x> := PolynomialRing(IntegerRing());
%o A153731 for n := 1 to 5 do
%o A153731     p := SwinnertonDyerPolynomial(n);
%o A153731     [c : c in Coefficients(p) | not IsZero(c)];
%o A153731 end for;  // _Peter Luschny_, Jun 12 2022
%o A153731 (Python) # See LINKS
%Y A153731 Cf. A247209, A354913 (left column).
%K A153731 sign,look,tabf
%O A153731 0,2
%A A153731 _Eric W. Weisstein_, Dec 31 2008
%E A153731 One term (row 0) prepended by _Alois P. Heinz_, Nov 28 2024