Real-rooted atlas

A053124: Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).

A053124

Proved for all n: all roots real, in (0, 1) · interlacing · alternating coefficient signs · basis map preserves real roots what this means

Recurrence from OEIS A053124
\(P_{n}(x) = \left(-2 + 4\,x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-242
33-16163
4-440-96644
55-80336-5122565
6-6140-8962304-256010246
77-2242016-768014080-1228840967
8-8336-403221120-5632079872-57344163848
99-4807392-50688183040-372736430080-262144655369
10-10660-12672109824-5125121397760-22937602228224-117964826214410
1111-88020592-2196481281280-44728329748480-1336934411206656-5242880104857611
12-121144-32032411840-292864012673024-3509452863504384-7471104055050240-23068672419430412
1313-145648048-7321606223360-32587776111132672-254017536392232960-403701760265289728-1006632961677721613
14-141820-698881244672-1244672077395968-317521920889061376-17258250242321285120-21223178241258291200-4362076166710886414
1515-224099008-203673623648768-171991040833495040-27941928966615662592-1114216857613264486400-109051904005888802816-187904819226843545615

Generated b-files: A000027 (row sums), A053124, A053125 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-3a7609bd7546c802 · revision 1 · JSON
Definition source
oeis_catalog:A053124
Fingerprint
f6-fd67808a966fe118

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:  -2,    4;
   3:   3,  -16,     16;
   4:  -4,   40,    -96,     64;
   5:   5,  -80,    336,   -512,     256;
   6:  -6,  140,   -896,   2304,   -2560,    1024;
   7:   7, -224,   2016,  -7680,   14080,  -12288,     4096;
   8:  -8,  336,  -4032,  21120,  -56320,   79872,   -57344,   16384;
   9:   9, -480,   7392, -50688,  183040, -372736,   430080, -262144,    65536;
  10: -10,  660, -12672, 109824, -512512, 1397760, -2293760, 2228224, -1179648, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(2-4*x)*P(n-1,x) - P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2 - 4*x)*P[n - 1, 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] = -2 + 4*x; for(n = 3, N, my(i = n); v[i] = -(2 - 4*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A053124: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:   1;
  1:  -2,    4;
  2:   3,  -16,     16;
  3:  -4,   40,    -96,     64;
  4:   5,  -80,    336,   -512,     256;
  5:  -6,  140,   -896,   2304,   -2560,    1024;
  6:   7, -224,   2016,  -7680,   14080,  -12288,     4096;
  7:  -8,  336,  -4032,  21120,  -56320,   79872,   -57344,   16384;
  8:   9, -480,   7392, -50688,  183040, -372736,   430080, -262144,    65536;
  9: -10,  660, -12672, 109824, -512512, 1397760, -2293760, 2228224, -1179648, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(2-4*x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2 - 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 - 4*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.

A053125: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:      1;
  1:      4,       -2;
  2:     16,      -16,       3;
  3:     64,      -96,      40,       -4;
  4:    256,     -512,     336,      -80,       5;
  5:   1024,    -2560,    2304,     -896,     140,      -6;
  6:   4096,   -12288,   14080,    -7680,    2016,    -224,      7;
  7:  16384,   -57344,   79872,   -56320,   21120,   -4032,    336,     -8;
  8:  65536,  -262144,  430080,  -372736,  183040,  -50688,   7392,   -480,   9;
  9: 262144, -1179648, 2228224, -2293760, 1397760, -512512, 109824, -12672, 660, -10;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(2-4*x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2 - 4*x)*P[n - 1, x] - 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] = -2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 - 4*x)*v[i - 1] - 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.

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-5f91bd7f8eb274ae
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-5f91bd7f8eb274ae, e-f83252f1a068273b
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f83252f1a068273b
  • 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-f83252f1a068273b
  • 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-9fb991ecf107efde
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [0, 1]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [0, 1]
    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-9fb991ecf107efde

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

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