Real-rooted atlas

A232535 → basis A007318 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(6 - 19\,n + n x + 19\,n^{2} - 3\,n^{2} x - 6\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-12 - 12\,x - 16\,n - 17\,n x - n x^{2} + 52\,n^{2} + 39\,n^{2} x - 7\,n^{2} x^{2} - 24\,n^{3} - 10\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2} - 8\,n x - 8\,n x^{2} + 12\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
214620
3807412166
4416592228201256
5204839362448540309002
697282355220352750410904262268
7450561315841459207641619040197456420046

All 15 rows

Technical details
Node id
n-4228dfcfa5518190 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9b4afe4f1acfaf5d

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:     14,       6;
  3:     80,      74,      12;
  4:    416,     592,     228,      20;
  5:   2048,    3936,    2448,     540,      30;
  6:   9728,   23552,   20352,    7504,    1090,     42;
  7:  45056,  131584,  145920,   76416,   19040,   1974,    56;
  8: 204800,  700416,  949248,  647680,  233280,  42336,  3304,   72;
  9: 917504, 3596288, 5763072, 4858880, 2309120, 614592, 85344, 5208, 90;
%F Formula
The row polynomials satisfy -(6*n^3-19*n^2+19*n-6-(2*n^3-3*n^2+n)*x)*P(n,x) = -(24*n^3-52*n^2+16*n+12+(10*n^3-39*n^2+17*n+12)*x-(6*n^3-7*n^2-n)*x^2)*P(n-1,x) + ((12*n^2-8*n-4)*x+(8*n^2-8*n-4)*x^2-4*n^2*x^3)*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[(-(24*n^3 - 52*n^2 + 16*n + 12 + (10*n^3 - 39*n^2 + 17*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*P[n - 1, x] + ((12*n^2 - 8*n - 4)*x + (8*n^2 - 8*n - 4)*x^2 - 4*n^2*x^3)*D[P[n - 1, x], x])/(-6*n^3 + 19*n^2 - 19*n + 6 + (2*n^3 - 3*n^2 + n)*x)]];
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] = (-(24*n^3 - 52*n^2 + 16*n + 12 + (10*n^3 - 39*n^2 + 17*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*v[i - 1] + ((12*n^2 - 8*n - 4)*x + (8*n^2 - 8*n - 4)*x^2 - 4*n^2*x^3)*deriv(v[i - 1], x))/(-6*n^3 + 19*n^2 - 19*n + 6 + (2*n^3 - 3*n^2 + n)*x)); 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