Real-rooted atlas

A108299 → D+1

Proved for all n: all roots real what this means

Recurrence
\(\left(-75 + 72\,x - 12\,x^{2} + 110\,n - 72\,n x + 8\,n x^{2} - 52\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-36 + 39\,x + 112\,x^{2} - 132\,x^{3} + 32\,x^{4} + 66\,n - 44\,n x - 180\,n x^{2} + 128\,n x^{3} - 16\,n x^{4} - 40\,n^{2} + 12\,n^{2} x + 96\,n^{2} x^{2} - 32\,n^{2} x^{3} + 8\,n^{3} - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-9\,x + 12\,x^{2} + 32\,x^{3} - 48\,x^{4} + 16\,x^{5} + 12\,n x - 8\,n x^{2} - 48\,n x^{3} + 32\,n x^{4} - 4\,n^{2} x + 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
20-1-1
30-3-1-4
40-511-3
50-73613
60-9515-2-18
70-11728-9-9-15
80-13945-20-3031-5

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-af9192a4a07453ac · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b9b953b2e3c32be8

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: 0,  -1;
   3: 0,  -3, -1;
   4: 0,  -5,  1,  1;
   5: 0,  -7,  3,  6,   1;
   6: 0,  -9,  5, 15,  -2,   -1;
   7: 0, -11,  7, 28,  -9,   -9, -1;
   8: 0, -13,  9, 45, -20,  -30,  3,  1;
   9: 0, -15, 11, 66, -35,  -70, 18, 12,  1;
  10: 0, -17, 13, 91, -54, -135, 50, 50, -4, -1;
%F Formula
The row polynomials satisfy (8*n^3-52*n^2+110*n-75+(16*n^2-72*n+72)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-40*n^2+66*n-36+(12*n^2-44*n+39)*x-(16*n^3-96*n^2+180*n-112)*x^2-(32*n^2-128*n+132)*x^3-(16*n-32)*x^4)*P(n-1,x) - ((4*n^2-12*n+9)*x+(8*n-12)*x^2-(16*n^2-48*n+32)*x^3-(32*n-48)*x^4-16*x^5)*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[((8*n^3 - 40*n^2 + 66*n - 36 + (12*n^2 - 44*n + 39)*x - (16*n^3 - 96*n^2 + 180*n - 112)*x^2 - (32*n^2 - 128*n + 132)*x^3 - (16*n - 32)*x^4)*P[n - 1, x] - ((4*n^2 - 12*n + 9)*x + (8*n - 12)*x^2 - (16*n^2 - 48*n + 32)*x^3 - (32*n - 48)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(8*n^3 - 52*n^2 + 110*n - 75 + (16*n^2 - 72*n + 72)*x + (8*n - 12)*x^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] = ((8*n^3 - 40*n^2 + 66*n - 36 + (12*n^2 - 44*n + 39)*x - (16*n^3 - 96*n^2 + 180*n - 112)*x^2 - (32*n^2 - 128*n + 132)*x^3 - (16*n - 32)*x^4)*v[i - 1] - ((4*n^2 - 12*n + 9)*x + (8*n - 12)*x^2 - (16*n^2 - 48*n + 32)*x^3 - (32*n - 48)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(8*n^3 - 52*n^2 + 110*n - 75 + (16*n^2 - 72*n + 72)*x + (8*n - 12)*x^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-b113ea8f5eac0e8f

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