Real-rooted atlas

basis-map coefficients of A145596 → D

Proved for all n: all roots real what this means

Recurrence
\(\left(-4 + 3\,n + n^{2}\right)\,P_{n}(x) = \left(10 - 2\,x + 2\,n + 4\,n x - 2\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-6 + 9\,x - 3\,x^{2} - 4\,n + 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2268
3-8161220
4-10-30602040
530-120-301603070
642126-58814035042112
7-112672-168-179284067256168

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-64e6cf669172ed21 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-f59a3fd2b0c3b9f1

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 ?

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:    0;
  1:    2;
  2:    2,     6;
  3:   -8,    16,   12;
  4:  -10,   -30,   60,    20;
  5:   30,  -120,  -30,   160,     30;
  6:   42,   126, -588,   140,    350,    42;
  7: -112,   672, -168, -1792,    840,   672,   56;
  8: -168,  -504, 4032, -3360,  -3780,  2772, 1176,   72;
  9:  420, -3360, 2520, 13440, -16800, -5040, 7140, 1920, 90;
%F Formula
The row polynomials satisfy (n^2+3*n-4)*P(n,x) = -(2*n^2-2*n-10-(3*n^2+4*n-2)*x)*P(n-1,x) - (4*n+6-(6*n+9)*x+(2*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 2*n - 10 - (3*n^2 + 4*n - 2)*x)*P[n - 1, x] - (4*n + 6 - (6*n + 9)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 3*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 2*n - 10 - (3*n^2 + 4*n - 2)*x)*v[i - 1] - (4*n + 6 - (6*n + 9)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 3*n - 4)); 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-01da8ccd2d2f1fb1

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023