Real-rooted atlas

A116414: Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

A116414 nonnegative coefficients

Proved for all n: all roots real, in (−8, 0) · interlacing · coefficients of one sign · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A116414
\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(-3\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
2138122
3404212195
412118487161409
53647314961482011760
610932736245410402252417573
732809844110646170188031828132585

Show all 15 stored rows

Generated b-files: A116414, A116415 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-6466a8550c08a52f · revision 1 · JSON
Definition source
oeis_catalog:A116414
Fingerprint
f6-5213fb95ee170aab

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:    13,      8,      1;
  3:    40,     42,     12,      1;
  4:   121,    184,     87,     16,     1;
  5:   364,    731,    496,    148,    20,     1;
  6:  1093,   2736,   2454,   1040,   225,    24,    1;
  7:  3280,   9844,  11064,   6170,  1880,   318,   28,   1;
  8:  9841,  34448,  46738,  32624, 13015,  3080,  427,  32,  1;
  9: 29524, 118101, 188208, 158724, 79044, 24381, 4704, 552, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) - 3*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] - 3*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] - 3*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.

A116414: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     4,      1;
  2:    13,      8,      1;
  3:    40,     42,     12,      1;
  4:   121,    184,     87,     16,     1;
  5:   364,    731,    496,    148,    20,     1;
  6:  1093,   2736,   2454,   1040,   225,    24,    1;
  7:  3280,   9844,  11064,   6170,  1880,   318,   28,   1;
  8:  9841,  34448,  46738,  32624, 13015,  3080,  427,  32,  1;
  9: 29524, 118101, 188208, 158724, 79044, 24381, 4704, 552, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) - 3*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] - 3*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] - 3*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-083de6f8f011746c
  • 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-083de6f8f011746c, e-7bedda0241d23786, e-894010de7c322dc8
  • 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-894010de7c322dc8
  • 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-7bedda0241d23786, e-894010de7c322dc8
  • 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-7bedda0241d23786, e-894010de7c322dc8
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-7bedda0241d23786
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-sqrt(3), sqrt(3)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-sqrt(3), sqrt(3)] 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-d9546d61239ca2e3
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-sqrt(3), sqrt(3)] to roots in [-4 - 2 sqrt(3), -4 + 2 sqrt(3)]: every real polynomial with all roots in E = [-sqrt(3), sqrt(3)] has an image with all roots in I = [-4 - 2 sqrt(3), -4 + 2 sqrt(3)]
    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-d9546d61239ca2e3
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-sqrt(3), 0] to roots in [-4 - 2 sqrt(3), -4 + 2 sqrt(3)]: every PF input with all roots in [-sqrt(3), 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-d9546d61239ca2e3

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