Real-rooted atlas

A099040 → x ↦ x−4

Proved for all n: all roots real, in (2, 4] · alternating coefficient signs · interlacing (weak) what this means

Recurrence
\(P_{n}(x) = \left(-8 + 2\,x\right)\,P_{n-1}(x) + \left(-8 + 2\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = -8 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-82-6
256-30430
3-384320-888-144
42624-29761252-23216684
5-1792025728-146244120-57632-3240
6122368-212608152480-5784812256-13766415336
7-8355841703936-1476608705536-20089634112-3200128-72576

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-7ca53e27e17e7a26 · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint
f6-a2e75800a0997b09

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 ?

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:        -8,         2;
  2:        56,       -30,          4;
  3:      -384,       320,        -88,        8;
  4:      2624,     -2976,       1252,     -232,        16;
  5:    -17920,     25728,     -14624,     4120,      -576,       32;
  6:    122368,   -212608,     152480,   -57848,     12256,    -1376,       64;
  7:   -835584,   1703936,   -1476608,   705536,   -200896,    34112,    -3200,    128;
  8:   5705728, -13357056,   13575680, -7829760,   2804496,  -639168,    90560,  -7296,    256;
  9: -38961152, 102965248, -120098816, 81191936, -35077248, 10047648, -1908992, 232064, -16384, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(8-2*x)*P(n-1,x) - (8-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -8 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -8 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(8 - 2*x)*P[n - 1, x] - (8 - 2*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] = -8 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 - 2*x)*v[i - 1] - (8 - 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 - 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 (2, 4]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-23d8b9b067cbe12f
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-23d8b9b067cbe12f
  • 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-23d8b9b067cbe12f

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