Real-rooted atlas

A108299 → D

Proved for all n: all roots real what this means

Recurrence
\(\left(12 - 72\,x - 14\,n + 108\,n x + 4\,n^{2} - 52\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(12 - 30\,x - 48\,x^{2} + 96\,x^{3} - 14\,n + 62\,n x + 52\,n x^{2} - 176\,n x^{3} + 4\,n^{2} - 40\,n^{2} x - 12\,n^{2} x^{2} + 96\,n^{2} x^{3} + 8\,n^{3} x - 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} - 12\,x^{3} + 32\,x^{4} - 2\,n x + 12\,n x^{2} + 8\,n x^{3} - 48\,n x^{4} - 4\,n^{2} x^{2} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = -1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
2-1-1
3-1-2-3
4-1-43-2
5-1-6643
6-1-8912-57
7-1-101224-15-64
8-1-121540-30-247-5

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-86eb0721424572d8 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-0962ebe4daaf5d6e

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 -2)
  • 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:  0;
   2: -1;
   3: -1,  -2;
   4: -1,  -4,  3;
   5: -1,  -6,  6,  4;
   6: -1,  -8,  9, 12,  -5;
   7: -1, -10, 12, 24, -15,   -6;
   8: -1, -12, 15, 40, -30,  -24,  7;
   9: -1, -14, 18, 60, -50,  -60, 28,  8;
  10: -1, -16, 21, 84, -75, -120, 70, 40, -9;
%F Formula
The row polynomials satisfy (4*n^2-14*n+12+(8*n^3-52*n^2+108*n-72)*x)*P(n,x) = (4*n^2-14*n+12+(8*n^3-40*n^2+62*n-30)*x-(12*n^2-52*n+48)*x^2-(16*n^3-96*n^2+176*n-96)*x^3)*P(n-1,x) - ((2*n-3)*x+(4*n^2-12*n+8)*x^2-(8*n-12)*x^3-(16*n^2-48*n+32)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := -1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 14*n + 12 + (8*n^3 - 40*n^2 + 62*n - 30)*x - (12*n^2 - 52*n + 48)*x^2 - (16*n^3 - 96*n^2 + 176*n - 96)*x^3)*P[n - 1, x] - ((2*n - 3)*x + (4*n^2 - 12*n + 8)*x^2 - (8*n - 12)*x^3 - (16*n^2 - 48*n + 32)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (8*n^3 - 52*n^2 + 108*n - 72)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = -1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 14*n + 12 + (8*n^3 - 40*n^2 + 62*n - 30)*x - (12*n^2 - 52*n + 48)*x^2 - (16*n^3 - 96*n^2 + 176*n - 96)*x^3)*v[i - 1] - ((2*n - 3)*x + (4*n^2 - 12*n + 8)*x^2 - (8*n - 12)*x^3 - (16*n^2 - 48*n + 32)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (8*n^3 - 52*n^2 + 108*n - 72)*x)); 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-df27c1125e074e5e

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023