Real-rooted atlas

A207607 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 11 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111819
27613232240
3521174810081283405
435712118022080646451253807
524476245516414912211008376322048935592
61677612777400718366456623361709056205824819217714233
71149851309769281184496481361080326192000012391424107724832768362105899

All 15 rows

Technical details
Node id
n-2a6cccf5866a7948 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-6231079da586bb57

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:       11,          8;
  2:       76,        132,          32;
  3:      521,       1748,        1008,         128;
  4:     3571,      21180,       22080,        6464,         512;
  5:    24476,     245516,      414912,      211008,       37632,        2048;
  6:   167761,    2777400,     7183664,     5662336,     1709056,      205824,       8192;
  7:  1149851,   30976928,   118449648,   136108032,    61920000,    12391424,    1077248,     32768;
  8:  7881196,  342568212,  1893468768,  3054188864,  1966883072,   582042624,   82952192,   5455872,   131072;
  9: 54018521, 3768798188, 29653854720, 65467355456, 57393146112, 23570291712, 4898611200, 522731520, 26935296, 524288;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 11 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 11 + 8*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 4*x)*P[n - 1, x] + 4*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] = 11 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 4*x)*v[i - 1] + 4*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