Real-rooted atlas

A124038: Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

A124038

Proved for all n: all roots real, in (−2, 3) · interlacing · basis map preserves real roots what this means

Recurrence from OEIS A124038
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = -2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1-21-1
2-1-21-2
32-2-21-1
414-3-211
5-236-4-212
6-1-668-5-211
72-4-121010-6-21-1
818-10-201512-7-21-2
9-2520-20-302114-8-21-1
10-1-101540-35-422816-9-211
112-6-303570-56-563618-10-212
12112-21-7070112-84-724520-11-211
13-2742-56-140126168-120-905522-12-21-1
14-1-1428112-126-252210240-165-1106624-13-21-2

Generated b-files: A124038 ?

Computed in your browser ?

Technical details
Node id
n-4ca7e225b5e9bcc6 · revision 1 · JSON
Definition source
oeis_catalog:A124038
Fingerprint
f6-17560ac5ecfb8518

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 ?

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1: -2,  1;
  2: -1, -2,   1;
  3:  2, -2,  -2,   1;
  4:  1,  4,  -3,  -2,   1;
  5: -2,  3,   6,  -4,  -2,  1;
  6: -1, -6,   6,   8,  -5, -2,  1;
  7:  2, -4, -12,  10,  10, -6, -2,  1;
  8:  1,  8, -10, -20,  15, 12, -7, -2,  1;
  9: -2,  5,  20, -20, -30, 21, 14, -8, -2, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, 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] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] - 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.

A124038: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1: -2,  1;
  2: -1, -2,   1;
  3:  2, -2,  -2,   1;
  4:  1,  4,  -3,  -2,   1;
  5: -2,  3,   6,  -4,  -2,  1;
  6: -1, -6,   6,   8,  -5, -2,  1;
  7:  2, -4, -12,  10,  10, -6, -2,  1;
  8:  1,  8, -10, -20,  15, 12, -7, -2,  1;
  9: -2,  5,  20, -20, -30, 21, 14, -8, -2, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-66ab36c44d35dd53
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-66ab36c44d35dd53, e-cf11db18e382a69a
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 3)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-cf11db18e382a69a
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1, 1] has a real-rooted image
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-c628eede86362da4

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 3 proof steps

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023