Real-rooted atlas

A199881 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-4}(x) + \left(-2\,x^{2}\right)\,P_{n-3}(x) + \left(2\,x - x^{2}\right)\,P_{n-2}(x) + \left(2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 4\,x + 3\,x^{2}\)
\(P_{3}(x) = 4\,x + 9\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21438
3049417
4021216535
50092825668
60032855367129
7000167096498239

All 15 rows

Technical details
Node id
n-56b5f636fe14e0c9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b0a9e6b200745e2b

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1, 2;
  2: 1, 4,  3;
  3: 0, 4,  9,  4;
  4: 0, 2, 12, 16,  5;
  5: 0, 0,  9, 28, 25,   6;
  6: 0, 0,  3, 28, 55,  36,   7;
  7: 0, 0,  0, 16, 70,  96,  49,   8;
  8: 0, 0,  0,  4, 55, 150, 154,  64,  9;
  9: 0, 0,  0,  0, 25, 150, 287, 232, 81, 10;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*x^2*P(n-3,x) + (2*x-x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + 3*x^2, P(3,x) = 4*x + 9*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 4*x + 3*x^2;
P[3, x_] := 4*x + 9*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*x^2*P[n - 3, x] + (2*x - x^2)*P[n - 2, x] + 2*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 4*x + 3*x^2; v[4] = 4*x + 9*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 2*x^2*v[i - 3] + (2*x - x^2)*v[i - 2] + 2*x*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.