Real-rooted atlas

A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.

A207607 nonnegative coefficients

Proved for all n: all roots real, in (−4, 0) · interlacing · coefficients of one sign · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A207607
\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21528
3199221
41142513255
51205549172144
612710514081212377
7135182336285121252987

Show all 15 stored rows

Generated b-files: A207607 ?

Computed in your browser ?

Technical details
Node id
n-353fb5a029e4e669 · revision 1 · JSON
Definition source
oeis_catalog:A207607
Fingerprint
f6-2b3b4ac7a26bccad

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): yes
  • 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: 1,  2;
  2: 1,  5,   2;
  3: 1,  9,   9,    2;
  4: 1, 14,  25,   13,    2;
  5: 1, 20,  55,   49,   17,    2;
  6: 1, 27, 105,  140,   81,   21,   2;
  7: 1, 35, 182,  336,  285,  121,  25,   2;
  8: 1, 44, 294,  714,  825,  506, 169,  29,  2;
  9: 1, 54, 450, 1386, 2079, 1716, 819, 225, 33, 2;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) - 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[(2 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 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.

A207607: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,   2;
   4: 1,  9,   9,    2;
   5: 1, 14,  25,   13,    2;
   6: 1, 20,  55,   49,   17,    2;
   7: 1, 27, 105,  140,   81,   21,   2;
   8: 1, 35, 182,  336,  285,  121,  25,   2;
   9: 1, 44, 294,  714,  825,  506, 169,  29,  2;
  10: 1, 54, 450, 1386, 2079, 1716, 819, 225, 33, 2;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) - 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[(2 + x)*P[n - 1, x] - P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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] = (2 + x)*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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 >= 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-4cbd4bb6f6420c24
  • 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-3cffc405672d7710, e-4cbd4bb6f6420c24, e-8f881604bf5a9d75
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-8f881604bf5a9d75
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n 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-3cffc405672d7710, e-8f881604bf5a9d75
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-3cffc405672d7710, e-8f881604bf5a9d75
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-3cffc405672d7710
  • 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-f2166ff27dacf1fa
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-4, 0]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-4, 0]
    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-f2166ff27dacf1fa
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 0] to roots in [-4, 0]: every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works
    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-f2166ff27dacf1fa

Proof chains

4 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 14 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