Real-rooted atlas

A049218 → D

Proved for all n all roots realinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(2\,n - 5\,n^{2} + 4\,n^{3} - n^{4}\right)\,P_{n-4}(x) + \left(-2\,n x + 2\,n^{2} x\right)\,P_{n-3}(x) + \left(-x^{2} - 2\,n^{2}\right)\,P_{n-2}(x) + \left(2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 2\,x\)
\(P_{2}(x) = -8 + 3\,x^{2}\)
\(P_{3}(x) = -40\,x + 4\,x^{3}\)
\(P_{4}(x) = 184 - 120\,x^{2} + 5\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
1022
2-803-5
30-4004-36
41840-1200569
5015680-280061294
6-8448073920-56007-1609
70-1047040255360-100808-80168
86485760-6892800722400-16800929865

All 15 rows

Technical details
Node id
n-5b50e616320aab8d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-205f5c838c09afd7

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,        2;
   2:        -8,        0,        3;
   3:         0,      -40,        0,        4;
   4:       184,        0,     -120,        0,         5;
   5:         0,     1568,        0,     -280,         0,      6;
   6:     -8448,        0,     7392,        0,      -560,      0,      7;
   7:         0,  -104704,        0,    25536,         0,  -1008,      0,     8;
   8:    648576,        0,  -689280,        0,     72240,      0,  -1680,     0,     9;
   9:         0, 10720512,        0, -3217280,         0, 177408,      0, -2640,     0, 10;
  10: -74972160,        0, 91901568,        0, -11968000,      0, 391776,     0, -3960,  0, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-4*n^3+5*n^2-2*n)*P(n-4,x) + (2*n^2-2*n)*x*P(n-3,x) - (2*n^2+x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 5, with P(1,x) = 2*x, P(2,x) = -8 + 3*x^2, P(3,x) = -40*x + 4*x^3, P(4,x) = 184 - 120*x^2 + 5*x^4.
%t Mathematica
P[1, x_] := 2*x;
P[2, x_] := -8 + 3*x^2;
P[3, x_] := -40*x + 4*x^3;
P[4, x_] := 184 - 120*x^2 + 5*x^4;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 4*n^3 + 5*n^2 - 2*n)*P[n - 4, x] + (2*n^2 - 2*n)*x*P[n - 3, x] - (2*n^2 + x^2)*P[n - 2, x] + 2*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2*x; v[2] = -8 + 3*x^2; v[3] = -40*x + 4*x^3; v[4] = 184 - 120*x^2 + 5*x^4; for(n = 5, N, my(i = n); v[i] = -(n^4 - 4*n^3 + 5*n^2 - 2*n)*v[i - 4] + (2*n^2 - 2*n)*x*v[i - 3] - (2*n^2 + x^2)*v[i - 2] + 2*x*v[i - 1]); 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