Real-rooted atlas

A113214 → x ↦ x−3

Proved for all n: all roots real, in (−1, 3] · interlacing (weak) what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-110
20-31-2
335-514
4-9-314-71-4
518-12-2527-910
6-275418-7144-1118
727-13563125-14965-131-16

Show all 15 stored rows

Generated b-files: A090132 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-87e7c3f73bb81aac · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint
f6-560ad1ce30eafb0f

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): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:  -1,    1;
  2:   0,   -3,    1;
  3:   3,    5,   -5,    1;
  4:  -9,   -3,   14,   -7,    1;
  5:  18,  -12,  -25,   27,   -9,    1;
  6: -27,   54,   18,  -71,   44,  -11,    1;
  7:  27, -135,   63,  125, -149,   65,  -13,   1;
  8:   0,  243, -324,  -81,  369, -267,   90, -15,   1;
  9: -81, -297,  891, -393, -616,  826, -433, 119, -17, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(3-x)*P(n-1,x) - (3-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[-(3 - x)*P[n - 1, x] - (3 - x)*P[n - 2, 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 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - x)*v[i - 1] - (3 - x)*v[i - 2]); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-1, 3]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f73e26dac50fbf3d
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f73e26dac50fbf3d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 1 proof step

No candidate equalities or related sequences.

References

  1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741