Real-rooted atlas

A039991 → x ↦ x+1

Proved for all n: all roots real what this means

Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 + 2\,x + x^{2} + n - 2\,n x - n x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 3\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
21-2-1-2
31-6-3-8
41-12-241-8
51-201020516
61-3045523-6-164
71-4211984-49-42-764

Show all 15 stored rows

Generated b-files: A128018 (row sums), A138230 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-e0e035772684cc42 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-06e89774407af090

Observed on the stored rows ?

  • Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (irregular)
  • 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 ?

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: 1;
  2: 1,  -2,  -1;
  3: 1,  -6,  -3;
  4: 1, -12,  -2,    4,    1;
  5: 1, -20,  10,   20,    5;
  6: 1, -30,  45,   52,    3,   -6,  -1;
  7: 1, -42, 119,   84,  -49,  -42,  -7;
  8: 1, -56, 252,   56, -250, -136,  -4,  8, 1;
  9: 1, -72, 468, -168, -738, -216, 132, 72, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1-(2*n-2)*x-(n-1)*x^2)*P(n-1,x) + (2*x+3*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 - (2*n - 2)*x - (n - 1)*x^2)*P[n - 1, x] + (2*x + 3*x^2 + x^3)*D[P[n - 1, x], x])/(n - 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 - (2*n - 2)*x - (n - 1)*x^2)*v[i - 1] + (2*x + 3*x^2 + x^3)*deriv(v[i - 1], x))/(n - 1)); 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-2abd4179002ed34f

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