Real-rooted atlas

A063967 → x ↦ x−6

Proved for all n: all roots real, in (1, 5] · alternating coefficient signs · interlacing (weak) what this means

Recurrence
\(P_{n}(x) = \left(-5 + x\right)\,P_{n-1}(x) + \left(-5 + x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -5 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-51-4
320-9112
4-7555-131-32
5275-285106-17180
6-10001350-695173-211-192
73625-60504010-1369256-251448
8-1312526125-212759295-2371355-291-1024

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-cc04537b3d998791 · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint
f6-93cd8b7280a9a4ce

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:      -5,       1;
   3:      20,      -9,       1;
   4:     -75,      55,     -13,      1;
   5:     275,    -285,     106,    -17,       1;
   6:   -1000,    1350,    -695,    173,     -21,     1;
   7:    3625,   -6050,    4010,  -1369,     256,   -25,     1;
   8:  -13125,   26125,  -21275,   9295,   -2371,   355,   -29,   1;
   9:   47500, -109875,  106400, -56895,   18501, -3765,   470, -33,   1;
  10: -171875,  453125, -509375, 323125, -128250, 33180, -5615, 601, -37, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(5-x)*P(n-1,x) - (5-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -5 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -5 + x;
P[n_, x_] := P[n, x] = Expand[-(5 - x)*P[n - 1, x] - (5 - 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] = -5 + x; for(n = 3, N, my(i = n); v[i] = -(5 - x)*v[i - 1] - (5 - 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 (1, 5]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-cc7a67409dceb768
  • 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-cc7a67409dceb768
  • 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-cc7a67409dceb768

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