Real-rooted atlas

A136668 → D

Proved for all n all roots realinterlacing (weak)

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(n^{2} - n^{3}\right)\,P_{n-4}(x) + \left(-8\,n^{2} x + 4\,n^{3} x\right)\,P_{n-3}(x) + \left(2 + n - 8\,n x^{2} - 2\,n^{2} + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 4\,x\)
\(P_{2}(x) = -11 + 24\,x^{2}\)
\(P_{3}(x) = -148\,x + 192\,x^{3}\)
\(P_{4}(x) = 119 - 1896\,x^{2} + 1920\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
1044
2-1102413
30-148019244
41190-189601920143
5032680-26432023040-124
6-14090720960-4099200322560-16673
70-65596015572480-707097605160960-418364
8188250-22231680348342400-134894592092897280-9367415

All 15 rows

Technical details
Node id
n-a17e506d0caf9445 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-419fcfe6085250cf

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 0)
  • 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:       0,       4;
   2:     -11,       0,       24;
   3:       0,    -148,        0,       192;
   4:     119,       0,    -1896,         0,        1920;
   5:       0,    3268,        0,    -26432,           0,     23040;
   6:   -1409,       0,    72096,         0,     -409920,         0,      322560;
   7:       0,  -65596,        0,   1557248,           0,  -7070976,           0,     5160960;
   8:   18825,       0, -2223168,         0,    34834240,         0,  -134894592,           0,     92897280;
   9:       0, 1333660,        0, -68928512,           0, 823129344,           0, -2826584064,            0, 1857945600;
  10: -283875,       0, 64464648,         0, -2106389440,         0, 20690191872,           0, -64620011520,          0, 40874803200;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n^3-n^2)*P(n-4,x) + (4*n^3-8*n^2)*x*P(n-3,x) - (2*n^2-n-2+(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (4*n^2-8*n)*x*P(n-1,x) for n >= 5, with P(1,x) = 4*x, P(2,x) = -11 + 24*x^2, P(3,x) = -148*x + 192*x^3, P(4,x) = 119 - 1896*x^2 + 1920*x^4.
%t Mathematica
P[1, x_] := 4*x;
P[2, x_] := -11 + 24*x^2;
P[3, x_] := -148*x + 192*x^3;
P[4, x_] := 119 - 1896*x^2 + 1920*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2)*P[n - 4, x] + (4*n^3 - 8*n^2)*x*P[n - 3, x] - (2*n^2 - n - 2 + (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (4*n^2 - 8*n)*x*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 4*x; v[2] = -11 + 24*x^2; v[3] = -148*x + 192*x^3; v[4] = 119 - 1896*x^2 + 1920*x^4; for(n = 5, N, my(i = n); v[i] = (-(n^3 - n^2)*v[i - 4] + (4*n^3 - 8*n^2)*x*v[i - 3] - (2*n^2 - n - 2 + (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (4*n^2 - 8*n)*x*v[i - 1])/(n - 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