Real-rooted atlas

A063967 → x ↦ x−7

Proved for all n: all roots real, in (2, 6] · alternating coefficient signs · interlacing (weak) what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-61-5
330-11120
4-14484-161-75
5684-552163-211275
6-32403348-1350267-261-1000
715336-193329918-2663396-3113625
8-72576108000-6739222944-4616550-361-13125

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-50a9a09f321fcea8 · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint
f6-4c21f592c13b2a9c

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,       1;
   3:       30,     -11,        1;
   4:     -144,      84,      -16,       1;
   5:      684,    -552,      163,     -21,       1;
   6:    -3240,    3348,    -1350,     267,     -26,     1;
   7:    15336,  -19332,     9918,   -2663,     396,   -31,      1;
   8:   -72576,  108000,   -67392,   22944,   -4616,   550,    -36,   1;
   9:   343440, -589248,   433512, -179160,   45601, -7334,    729, -41,   1;
  10: -1625184, 3158352, -2677968, 1303416, -402126, 81689, -10942, 933, -46, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(6-x)*P(n-1,x) - (6-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -6 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -6 + x;
P[n_, x_] := P[n, x] = Expand[-(6 - x)*P[n - 1, x] - (6 - 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 + x; for(n = 3, N, my(i = n); v[i] = -(6 - x)*v[i - 1] - (6 - 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, 6]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-12ad66072da87843
  • 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-12ad66072da87843
  • 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-12ad66072da87843

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