Real-rooted atlas

A113953 → x ↦ x−15

Proved for all n: all roots real, in (7, 15] · alternating coefficient signs · interlacing (weak) what this means

Recurrence
\(P_{n}(x) = \left(-15 + x\right)\,P_{n-1}(x) + \left(-30 + 2\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = -15 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-151-14
2195-281168
3-2475555-411-1960
431275-95701084-54122736
5-394875153225-234901782-671-263424
64984875-2343600453915-464322649-8013051328
7-6292687534752375-81411751049955-805933685-931-35342720

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-eb460955a5ddb26b · revision 1 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint
f6-62a4b2b77a02531a

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:          -15,          1;
  2:          195,        -28,           1;
  3:        -2475,        555,         -41,         1;
  4:        31275,      -9570,        1084,       -54,         1;
  5:      -394875,     153225,      -23490,      1782,       -67,       1;
  6:      4984875,   -2343600,      453915,    -46432,      2649,     -80,       1;
  7:    -62926875,   34752375,    -8141175,   1049955,    -80593,    3685,     -93,    1;
  8:    794356875, -503934750,   138565350, -21589710,   2086516, -128170,    4890, -106,    1;
  9: -10027546875, 7184953125, -2268675000, 414630000, -48369750, 3737330, -191360, 6264, -119, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(15-x)*P(n-1,x) - (30-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -15 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -15 + x;
P[n_, x_] := P[n, x] = Expand[-(15 - x)*P[n - 1, x] - (30 - 2*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -15 + x; for(n = 2, N, my(i = n + 1); v[i] = -(15 - x)*v[i - 1] - (30 - 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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (7, 15]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0ffa0091fcab81ab
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-0ffa0091fcab81ab
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-0ffa0091fcab81ab

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