Real-rooted atlas

A099097 → x ↦ x−2

Proved for all n: all roots real, in (14/9, 2] · alternating coefficient signs · interlacing (weak) what this means

Recurrence
\(P_{n}(x) = \left(-6 + 3\,x\right)\,P_{n-1}(x) + \left(-2 + x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-63-3
334-3598
4-192300-15627-21
51084-22721783-6218155
6-612016092-169028865-2322243-144
734552-109284143850-10087139744-8343729377
8-195072721056-11410561002144-527568166482-291602187-987

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-0a274609716903e8 · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint
f6-fbaeed9a78101bba

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 ?

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:       -6,        3;
   3:       34,      -35,         9;
   4:     -192,      300,      -156,       27;
   5:     1084,    -2272,      1783,     -621,        81;
   6:    -6120,    16092,    -16902,     8865,     -2322,      243;
   7:    34552,  -109284,    143850,  -100871,     39744,    -8343,       729;
   8:  -195072,   721056,  -1141056,  1002144,   -527568,   166482,    -29160,    2187;
   9:  1101328, -4658432,   8612520, -9090440,   5991481, -2525166,    664605,  -99873,    6561;
  10: -6217824, 29617392, -62647248, 77234856, -61162926, 32264907, -11338326, 2559519, -336798, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(6-3*x)*P(n-1,x) - (2-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -6 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(6 - 3*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] = -6 + 3*x; for(n = 3, N, my(i = n); v[i] = -(6 - 3*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 (14/9, 2]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-fbe1f1ba82c385a7
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-fbe1f1ba82c385a7
  • 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-fbe1f1ba82c385a7

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