Real-rooted atlas

A063967 → x ↦ x−4

Proved for all n: all roots real, in (−1, 3] · interlacing (weak) what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-31-2
36-512
4-915-710
59-3328-91-4
6054-8145-1118
7-27-54180-16166-131-8
881-27-297447-28191-1510

Show all 15 stored rows

Generated b-files: A108520 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-2f435ee30c76fa47 · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint
f6-9c3db5b54ab0cc16

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:   -3,    1;
   3:    6,   -5,    1;
   4:   -9,   15,   -7,    1;
   5:    9,  -33,   28,   -9,     1;
   6:    0,   54,  -81,   45,   -11,    1;
   7:  -27,  -54,  180, -161,    66,  -13,    1;
   8:   81,  -27, -297,  447,  -281,   91,  -15,   1;
   9: -162,  297,  270, -975,   931, -449,  120, -17,   1;
  10:  243, -891,  351, 1557, -2478, 1724, -673, 153, -19, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(3-x)*P(n-1,x) - (3-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -3 + x;
P[n_, x_] := P[n, x] = Expand[-(3 - x)*P[n - 1, x] - (3 - 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] = -3 + x; for(n = 3, N, my(i = n); v[i] = -(3 - x)*v[i - 1] - (3 - 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, 3]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0e7024e7b55599e9
  • 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-0e7024e7b55599e9

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