Real-rooted atlas

A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

A207823 nonnegative coefficients

Proved for all n: all roots real, in (−6, −2) · interlacing · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A207823
\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
2158124
35646121115
420923293161551
578010915921562012640
6291149123366120023524112649
71086421468177848010212033028160605

Show all 15 stored rows

Generated b-files: A207823 ?

Computed in your browser ?

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(2\,n\right)\,P_{n}(x) = \left(4 + x + 4\,n + n x\right)\,P_{n-1}(x) + \left(12 + 8\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-30cb3cad216ed189 · revision 3 · JSON
Definition source
oeis_catalog:A207823
derived:basis_map_A007318 (equal by e-a3cabaafc7100148)
derived:shift_x_plus_one (equal by e-2a623f36080b1454)
Fingerprint
f6-07a6a8e112979ca6

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:      4,      1;
  2:     15,      8,      1;
  3:     56,     46,     12,      1;
  4:    209,    232,     93,     16,      1;
  5:    780,   1091,    592,    156,     20,     1;
  6:   2911,   4912,   3366,   1200,    235,    24,    1;
  7:  10864,  21468,  17784,   8010,   2120,   330,   28,   1;
  8:  40545,  91824,  89238,  48624,  16255,  3416,  441,  32,  1;
  9: 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(4 + 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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A207823: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,      1;
  2:     15,      8,      1;
  3:     56,     46,     12,      1;
  4:    209,    232,     93,     16,      1;
  5:    780,   1091,    592,    156,     20,     1;
  6:   2911,   4912,   3366,   1200,    235,    24,    1;
  7:  10864,  21468,  17784,   8010,   2120,   330,   28,   1;
  8:  40545,  91824,  89238,  48624,  16255,  3416,  441,  32,  1;
  9: 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (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.

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-7d81269ae54a0979
  • 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-720cd5df5ac0e3b3, e-7d81269ae54a0979, e-ec629e395a3def0c
  • 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-720cd5df5ac0e3b3
  • 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-56c0c02612a8191c, e-720cd5df5ac0e3b3, e-9c2f7b8109ebb5e1, e-ec629e395a3def0c
  • 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-56c0c02612a8191c, e-720cd5df5ac0e3b3, e-9c2f7b8109ebb5e1, e-ec629e395a3def0c
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, -2)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-56c0c02612a8191c, e-ec629e395a3def0c
  • 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-013b61d0172203f3
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-6, -2]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-6, -2]
    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-013b61d0172203f3
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 0] to roots in [-6, -2]: 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-013b61d0172203f3
  • 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-9c2f7b8109ebb5e1
  • 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-56c0c02612a8191c, e-9c2f7b8109ebb5e1

Proof chains

8 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