Real-rooted atlas

A053122 → x ↦ x+1 → D+1

Proved for all n all roots realinterlacing (weak)

Recurrence

\(\left(-10\,x + 2\,x^{2} + 4\,n + 10\,n x - 2\,n x^{2} - 6\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(4\,n - 4\,n x + 5\,n x^{2} - n x^{3} - n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-6 - 13\,x - x^{2} + 5\,x^{3} - x^{4} + 3\,n + 2\,n x - n x^{2} - 3\,n^{2} - 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
3-201-1
42-501-2
518-9011
6-4020-14013
73-16-940-2001-1
8221-36-3570-2701-4

All 15 rows

Technical details
Node id
n-7f8a59ba04fc7524 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-74ec4878e96f8238

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  0,   1;
   3: -2,   0,   1;
   4:  2,  -5,   0,   1;
   5:  1,   8,  -9,   0,   1;
   6: -4,   0,  20, -14,   0,    1;
   7:  3, -16,  -9,  40, -20,    0,   1;
   8:  2,  21, -36, -35,  70,  -27,   0,   1;
   9: -6,   0,  75, -56, -90,  112, -35,   0, 1;
  10:  4, -33, -36, 190, -56, -189, 168, -44, 0, 1;
%F Formula
The row polynomials satisfy (2*n^3-6*n^2+4*n+(10*n-10)*x-(2*n-2)*x^2)*P(n,x) = -(n^3-4*n-(n^3-4*n)*x-5*n*x^2+n*x^3)*P(n-1,x) - (3*n^2-3*n+6+(2*n^2-2*n+13)*x-(n^2-n-1)*x^2-5*x^3+x^4)*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[(-(n^3 - 4*n - (n^3 - 4*n)*x - 5*n*x^2 + n*x^3)*P[n - 1, x] - (3*n^2 - 3*n + 6 + (2*n^2 - 2*n + 13)*x - (n^2 - n - 1)*x^2 - 5*x^3 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 4*n + (10*n - 10)*x - (2*n - 2)*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] = (-(n^3 - 4*n - (n^3 - 4*n)*x - 5*n*x^2 + n*x^3)*v[i - 1] - (3*n^2 - 3*n + 6 + (2*n^2 - 2*n + 13)*x - (n^2 - n - 1)*x^2 - 5*x^3 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 4*n + (10*n - 10)*x - (2*n - 2)*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)

polytool c48629374597

Properties

Proof chains ? JSON

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