Real-rooted atlas

A063967 → x ↦ x−3

Proved for all n: all roots real, in (−2, 2] · interlacing (weak) what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-21-1
32-310
404-411
5-407-51-1
68-12-211-610
7-828-22-716-711
80-3264-32-1622-81-1
9160-88120-39-3029-910
10-328016-200198-39-5037-1011
1132-17622488-398297-27-7746-111-1
120192-576464288-7164123-11256-1210
13-640720-1456772728-118953358-15667-1311
14128-448-962128-311210361566-1849644146-21079-141-1
15-128960-1696-7205352-586810103009-2720722276-27592-1510

Computed in your browser ?

Technical details
Node id
n-95ccc13d8da74c1c · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint
f6-b11f45bf83eef983

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 ?

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:  -2,   1;
   3:   2,  -3,   1;
   4:   0,   4,  -4,    1;
   5:  -4,   0,   7,   -5,   1;
   6:   8, -12,  -2,   11,  -6,   1;
   7:  -8,  28, -22,   -7,  16,  -7,   1;
   8:   0, -32,  64,  -32, -16,  22,  -8,  1;
   9:  16,   0, -88,  120, -39, -30,  29, -9,   1;
  10: -32,  80,  16, -200, 198, -39, -50, 37, -10, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(2-x)*P(n-1,x) - (2-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 1, x] - (2 - 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] = -2 + x; for(n = 3, N, my(i = n); v[i] = -(2 - x)*v[i - 1] - (2 - 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, 2]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-acc0c42734fdb8fc
  • 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-acc0c42734fdb8fc

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