Real-rooted atlas

A053123 → x ↦ x+1

Proved for all n: all roots real what this means

Recurrence
\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(6 + 8\,x - 4\,n - 6\,n x\right)\,P_{n-1}(x) + \left(3 + 7\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-1-2-3
30235
412-2-4-3
5-1-6-905-11
60418245-645
714-4-36-49-147-91
8-1-10-30-20508428-893

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-9b98aa08088dc145 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b676e85961482bdd

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 -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:  1;
   2: -1,  -2;
   3:  0,   2,   3;
   4:  1,   2,  -2,   -4;
   5: -1,  -6,  -9,    0,    5;
   6:  0,   4,  18,   24,    5,   -6;
   7:  1,   4,  -4,  -36,  -49,  -14,    7;
   8: -1, -10, -30,  -20,   50,   84,   28,  -8;
   9:  0,   6,  45,  120,  115,  -36, -126, -48,  9;
  10:  1,   6,  -6, -120, -335, -358,  -48, 168, 75, -10;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = -(4*n-6+(6*n-8)*x)*P(n-1,x) + (3+7*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 6 + (6*n - 8)*x)*P[n - 1, x] + (3 + 7*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n - 6 + (6*n - 8)*x)*v[i - 1] + (3 + 7*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 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-fb33150e82d07f76

Proof chains

1 incoming proof step. 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