Real-rooted atlas

A091044 → basis A007318 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 - 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-x - 4\,n - 7\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
216622
3968012188
4512672240201444
52560460826885603010446
6122882816023040806411204272714
7573441597441689608448020160201656492760

All 15 rows

Technical details
Node id
n-68a5093a060a8a72 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3ec9a58a3a187a12

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       2;
  2:      16,       6;
  3:      96,      80,      12;
  4:     512,     672,     240,      20;
  5:    2560,    4608,    2688,     560,      30;
  6:   12288,   28160,   23040,    8064,    1120,     42;
  7:   57344,  159744,  168960,   84480,   20160,   2016,    56;
  8:  262144,  860160, 1118208,  732160,  253440,  44352,  3360,   72;
  9: 1179648, 4456448, 6881280, 5591040, 2562560, 658944, 88704, 5280, 90;
%F Formula
The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (8*n^2-4*n+(6*n^2-7*n-1)*x)*P(n-1,x) - (4*n*x+4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (6*n^2 - 7*n - 1)*x)*P[n - 1, x] - (4*n*x + 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 4*n + (6*n^2 - 7*n - 1)*x)*v[i - 1] - (4*n*x + 4*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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