Real-rooted atlas

A053124 → D+1

Proved for all n all roots realinterlacing (weak)

Recurrence

\(\left(3 - 8\,x + 2\,x^{2} + n + 8\,n x - 2\,n x^{2} - 6\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(5\,n - 14\,n x + 14\,n x^{2} - 4\,n x^{3} - 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 10\,x^{2} + 12\,x^{3} - 4\,x^{4} + 4\,n x - 4\,n x^{2} - 4\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
3-13161619
436-152966444
5-75592-120051225685
6134-16526016-793625601024146
7-2173808-2102448640-47360122884096231
8328-772859328-204160343040-2641925734416384344

All 15 rows

Technical details
Node id
n-2119584ac59ad687 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5bbe1bfb388d2201

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:    2,      4;
   3:  -13,     16,      16;
   4:   36,   -152,      96,       64;
   5:  -75,    592,   -1200,      512,      256;
   6:  134,  -1652,    6016,    -7936,     2560,      1024;
   7: -217,   3808,  -21024,    48640,   -47360,     12288,     4096;
   8:  328,  -7728,   59328,  -204160,   343040,   -264192,    57344,    16384;
   9: -471,  14304, -144672,   681472, -1680640,   2207744, -1404928,   262144,   65536;
  10:  650, -24684,  316800, -1940224,  6476288, -12364800, 13303808, -7208960, 1179648, 262144;
%F Formula
The row polynomials satisfy (2*n^3-6*n^2+n+3+(8*n-8)*x-(2*n-2)*x^2)*P(n,x) = -(2*n^3-5*n-(4*n^3-14*n)*x-14*n*x^2+4*n*x^3)*P(n-1,x) - ((4*n^2-4*n-2)*x-(4*n^2-4*n-10)*x^2-12*x^3+4*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[(-(2*n^3 - 5*n - (4*n^3 - 14*n)*x - 14*n*x^2 + 4*n*x^3)*P[n - 1, x] - ((4*n^2 - 4*n - 2)*x - (4*n^2 - 4*n - 10)*x^2 - 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + n + 3 + (8*n - 8)*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] = (-(2*n^3 - 5*n - (4*n^3 - 14*n)*x - 14*n*x^2 + 4*n*x^3)*v[i - 1] - ((4*n^2 - 4*n - 2)*x - (4*n^2 - 4*n - 10)*x^2 - 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + n + 3 + (8*n - 8)*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. [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