Real-rooted atlas

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

A207606 nonnegative coefficients

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

Recurrence from OEIS A207606
\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 4 + 7\,x + 2\,x^{2}\)
\(P_{1}(x) = 5 + 16\,x + 11\,x^{2} + 2\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
047213
151611234
26303615289
37509164192233
4877196204100232610
591123785403851442721597
610156672125412106501963124181
7112101122264032892366101525635210946

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-dfab35176bd08e72 · revision 1 · JSON
Definition source
oeis_catalog:A207606
Fingerprint
f6-c538a4320031e08b

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 ?

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:  4,   7,    2;
  1:  5,  16,   11,    2;
  2:  6,  30,   36,   15,     2;
  3:  7,  50,   91,   64,    19,     2;
  4:  8,  77,  196,  204,   100,    23,     2;
  5:  9, 112,  378,  540,   385,   144,    27,    2;
  6: 10, 156,  672, 1254,  1210,   650,   196,   31,    2;
  7: 11, 210, 1122, 2640,  3289,  2366,  1015,  256,   35,   2;
  8: 12, 275, 1782, 5148,  8008,  7371,  4200, 1496,  324,  39,  2;
  9: 13, 352, 2717, 9438, 17875, 20384, 14756, 6936, 2109, 400, 43, 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) = 4 + 7*x + 2*x^2, P(1,x) = 5 + 16*x + 11*x^2 + 2*x^3.
%t Mathematica
P[0, x_] := 4 + 7*x + 2*x^2;
P[1, x_] := 5 + 16*x + 11*x^2 + 2*x^3;
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] = 4 + 7*x + 2*x^2; v[2] = 5 + 16*x + 11*x^2 + 2*x^3; 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.

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-ad1c1146bac40fcc
  • 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-3222de37fcca30b3, e-62ac8b7225a515c7, e-ad1c1146bac40fcc
  • 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-62ac8b7225a515c7
  • 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-3222de37fcca30b3, e-62ac8b7225a515c7
  • 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-3222de37fcca30b3, e-62ac8b7225a515c7
  • 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-3222de37fcca30b3

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

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