Real-rooted atlas

A053123: Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).

A053123

Proved for all n: all roots real what this means

Recurrence from OEIS A053123
\(P_{n}(x) = \left(1 - 2\,x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 - 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
21-2-1
31-430
41-610-41
51-821-205-1
61-1036-5635-60
71-1255-120126-5671
81-1478-220330-25284-8-1

Show all 15 stored rows

Generated b-files: A053122 (reversed triangle), A053123 ?

Computed in your browser ?

Technical details
Node id
n-0b6e037cc63716ca · revision 1 · JSON
Definition source
oeis_catalog:A053123
Fingerprint
f6-63d46ebead6c4d0e

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: 1,  -2;
   3: 1,  -4,   3;
   4: 1,  -6,  10,   -4;
   5: 1,  -8,  21,  -20,    5;
   6: 1, -10,  36,  -56,   35,    -6;
   7: 1, -12,  55, -120,  126,   -56,    7;
   8: 1, -14,  78, -220,  330,  -252,   84,   -8;
   9: 1, -16, 105, -364,  715,  -792,  462, -120,   9;
  10: 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10;
%F Formula
The row polynomials satisfy P(n,x) = (1-2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 - 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 - 2*x;
P[n_, x_] := P[n, x] = Expand[(1 - 2*x)*P[n - 1, x] - x^2*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] = 1 - 2*x; for(n = 3, N, my(i = n); v[i] = (1 - 2*x)*v[i - 1] - x^2*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.

A053122: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:   1;
  1:  -2,    1;
  2:   3,   -4,    1;
  3:  -4,   10,   -6,    1;
  4:   5,  -20,   21,   -8,     1;
  5:  -6,   35,  -56,   36,   -10,    1;
  6:   7,  -56,  126, -120,    55,  -12,    1;
  7:  -8,   84, -252,  330,  -220,   78,  -14,   1;
  8:   9, -120,  462, -792,   715, -364,  105, -16,   1;
  9: -10,  165, -792, 1716, -2002, 1365, -560, 136, -18, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1-2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 - 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 - 2*x;
P[n_, x_] := P[n, x] = Expand[(1 - 2*x)*P[n - 1, x] - x^2*P[n - 2, x]];
Table[Reverse[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] = 1 - 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 - 2*x)*v[i - 1] - x^2*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A053123: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  -2;
  2: 1,  -4,   3;
  3: 1,  -6,  10,   -4;
  4: 1,  -8,  21,  -20,    5;
  5: 1, -10,  36,  -56,   35,    -6;
  6: 1, -12,  55, -120,  126,   -56,    7;
  7: 1, -14,  78, -220,  330,  -252,   84,   -8;
  8: 1, -16, 105, -364,  715,  -792,  462, -120,   9;
  9: 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10;
%F Formula
The row polynomials satisfy P(n,x) = (1-2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 - 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 - 2*x;
P[n_, x_] := P[n, x] = Expand[(1 - 2*x)*P[n - 1, x] - x^2*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] = 1 - 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 - 2*x)*v[i - 1] - x^2*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 >= 1, 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-b3aa6b8d6262c436

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 3 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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