Real-rooted atlas

A091044 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 27\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(-27\,x - 18\,x^{2}\right)\,P_{n-2}'(x) + \left(-9\,x^{2}\right)\,P_{n-2}''(x) + \left(10 + 6\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 10 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110616
29112927247
3820212410081084060
473813180326109621040571908
5664304545305806082323623321014581368598
659787163268651188816371644501694196161595510327838243
7538084086660376231419808197804376690703201077753673483217496601865584

All 15 rows

Technical details
Node id
n-3c978d7ddc5e48c2 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-7e23844fe6099114

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 0)
  • 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:         1;
  1:        10,           6;
  2:        91,         129,          27;
  3:       820,        2124,        1008,          108;
  4:      7381,       31803,       26109,         6210,         405;
  5:     66430,      454530,      580608,       232362,       33210,        1458;
  6:    597871,     6326865,    11888163,      7164450,     1694196,      161595,       5103;
  7:   5380840,    86660376,   231419808,    197804376,    69070320,    10777536,     734832,     17496;
  8:  48427561,  1174941399,  4358167065,   5089464522,  2466709119,   560961126,   62062686,   3175524,    59049;
  9: 435848050, 15824571534, 80236174896, 124810787574, 80759142450, 25128577026, 4013984808, 331225524, 13187610, 196830;
%F Formula
The row polynomials satisfy P(n,x) = -(9+27*x+9*x^2)*P(n-2,x) - (27*x+18*x^2)*d/dx P(n-2,x) - 9*x^2*d^2/dx^2 P(n-2,x) + (10+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 10 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 27*x + 9*x^2)*P[n - 2, x] - (27*x + 18*x^2)*D[P[n - 2, x], x] - 9*x^2*D[P[n - 2, x], {x, 2}] + (10 + 6*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 27*x + 9*x^2)*v[i - 2] - (27*x + 18*x^2)*deriv(v[i - 2], x) - 9*x^2*deriv(deriv(v[i - 2], x), x) + (10 + 6*x)*v[i - 1] + 6*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023