Real-rooted atlas

A063967 → x ↦ x−2

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-110
30-110
41-1-110
5-13-2-110
60-25-3-110
71-2-27-4-110
8-15-7-19-5-110

Show all 15 stored rows

Generated b-files: A199324 ?

Computed in your browser ?

Technical details
Node id
n-500943bf802dcc95 · revision 2 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A026729
Fingerprint
f6-21068b267380f889

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2: -1,  1;
   3:  0, -1,  1;
   4:  1, -1, -1,   1;
   5: -1,  3, -2,  -1,   1;
   6:  0, -2,  5,  -3,  -1,  1;
   7:  1, -2, -2,   7,  -4, -1,  1;
   8: -1,  5, -7,  -1,   9, -5, -1,  1;
   9:  0, -3, 12, -15,   1, 11, -6, -1,  1;
  10:  1, -3, -3,  21, -26,  4, 13, -7, -1, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) - (1-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 + x; for(n = 3, N, my(i = n); v[i] = -(1 - x)*v[i - 1] - (1 - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A199324: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1: -1,  1;
  2:  0, -1,  1;
  3:  1, -1, -1,   1;
  4: -1,  3, -2,  -1,   1;
  5:  0, -2,  5,  -3,  -1,  1;
  6:  1, -2, -2,   7,  -4, -1,  1;
  7: -1,  5, -7,  -1,   9, -5, -1,  1;
  8:  0, -3, 12, -15,   1, 11, -6, -1,  1;
  9:  1, -3, -3,  21, -26,  4, 13, -7, -1, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) - (1-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[-(1 - x)*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = -(1 - x)*v[i - 1] - (1 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 1, is zero or real-rooted with all roots in (-3, 1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-1365676e8cb0fd13, e-3c4e27dedb9c54a8
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-1365676e8cb0fd13, e-3c4e27dedb9c54a8

Proof chains

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

Used in 2 proof steps

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