Real-rooted atlas

Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.

A136668 (tail)

Proved for all n: all roots real, in (−2, 2) · interlacing what this means

Recurrence from OEIS A136668
\(P_{n}(x) = \left(2\,n x\right)\,P_{n-1}(x) + \left(-1 - n\right)\,P_{n-2}(x)\)
\(P_{1}(x) = -2 + 2\,x^{2}\)
\(P_{2}(x) = -11\,x + 8\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1-2020
20-1108-3
380-74048-18
401190-6320384-129
5-48016340-660803840-1182
60-14090240320-81984046080-13281
73840-3279803893120-11784960645120-176478
80188250-741056069668480-19270656010321920-2704119

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-1eae5c7270ad0739 · revision 1 · JSON
Definition source
oeis_catalog:A136668
Fingerprint
f6-f01cd712f19a05d6

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 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 ?

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    -2,       0,      2;
   2:     0,     -11,      0,        8;
   3:     8,       0,    -74,        0,        48;
   4:     0,     119,      0,     -632,         0,        384;
   5:   -48,       0,   1634,        0,     -6608,          0,      3840;
   6:     0,   -1409,      0,    24032,         0,     -81984,         0,      46080;
   7:   384,       0, -32798,        0,    389312,          0,  -1178496,          0,     645120;
   8:     0,   18825,      0,  -741056,         0,    6966848,         0,  -19270656,          0,    10321920;
   9: -3840,       0, 666830,        0, -17232128,          0, 137188224,          0, -353323008,           0, 185794560;
  10:     0, -283875,      0, 21488216,         0, -421277888,         0, 2955741696,          0, -7180001280,         0, 3715891200;
%F Formula
The row polynomials satisfy P(n,x) = 2*n*x*P(n-1,x) - (n+1)*P(n-2,x) for n >= 3, with P(1,x) = -2 + 2*x^2, P(2,x) = -11*x + 8*x^3.
%t Mathematica
P[1, x_] := -2 + 2*x^2;
P[2, x_] := -11*x + 8*x^3;
P[n_, x_] := P[n, x] = Expand[2*n*x*P[n - 1, x] - (n + 1)*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] = -2 + 2*x^2; v[2] = -11*x + 8*x^3; for(n = 3, N, my(i = n); v[i] = 2*n*x*v[i - 1] - (n + 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.

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-65533d28bec86ec5
  • 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-65533d28bec86ec5, e-f1f9c50c96bd4da9
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, 2)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f1f9c50c96bd4da9

Proof chains

2 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