Real-rooted atlas

A063967 → x ↦ x−10

Proved for all n: all roots real, in (5, 9] · alternating coefficient signs · interlacing (weak) what this means

Recurrence
\(P_{n}(x) = \left(-9 + x\right)\,P_{n-1}(x) + \left(-9 + x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -9 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-91-8
372-17156
4-567207-251-384
54455-2205406-3312624
6-3499221870-5427669-411-17920
7274833-20752264854-10745996-491122368
8-21585691910709-720495150111-186711387-571-835584

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-2445c52c3b45c0fd · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint
f6-31cf3f3506415c25

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:         -9,         1;
   3:         72,       -17,         1;
   4:       -567,       207,       -25,        1;
   5:       4455,     -2205,       406,      -33,        1;
   6:     -34992,     21870,     -5427,      669,      -41,      1;
   7:     274833,   -207522,     64854,   -10745,      996,    -49,      1;
   8:   -2158569,   1910709,   -720495,   150111,   -18671,   1387,    -57,    1;
   9:   16953624, -17212419,   7603956, -1909935,   298441, -29717,   1842,  -65,   1;
  10: -133155495, 152510445, -77252859, 22721877, -4277754, 534740, -44395, 2361, -73, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(9-x)*P(n-1,x) - (9-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -9 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -9 + x;
P[n_, x_] := P[n, x] = Expand[-(9 - x)*P[n - 1, x] - (9 - 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] = -9 + x; for(n = 3, N, my(i = n); v[i] = -(9 - x)*v[i - 1] - (9 - 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 (5, 9]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-25462190c1ff20a9
  • 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-25462190c1ff20a9
  • 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-25462190c1ff20a9

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