Real-rooted atlas

A207606 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 26 + 17\,x + 2\,x^{2}\)
\(P_{1}(x) = 97 + 109\,x + 29\,x^{2} + 2\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02617245
197109292237
23626252814321313
31351337822825805927652
4504217616167896299105077246875
5188178974411606860302146401741972301411
67022644992176936353188317637230232270911922030827
72620872230087495003144322421928219446719572254016143214310771

All 15 rows

Technical details
Node id
n-302c443e9dec0901 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-19d8631da43cf9f4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      26,       17,         2;
  1:      97,      109,        29,         2;
  2:     362,      625,       281,        43,         2;
  3:    1351,     3378,      2282,       580,        59,        2;
  4:    5042,    17616,     16789,      6299,      1050,       77,        2;
  5:   18817,    89744,    116068,     60302,     14640,     1741,       97,       2;
  6:   70226,   449921,    769363,    531883,    176372,    30232,     2709,     119,      2;
  7:  262087,  2230087,   4950031,   4432242,   1928219,   446719,    57225,    4016,    143,    2;
  8:  978122, 10962601,  31160910,  35443842,  19681622,  5918458,  1016260,  101282,   5730,  169,   2;
  9: 3650401, 53561040, 192978030, 274835562, 190968599, 72501025, 16023833, 2126346, 169899, 7925, 197, 2;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 26 + 17*x + 2*x^2, P(1,x) = 97 + 109*x + 29*x^2 + 2*x^3.
%t Mathematica
P[0, x_] := 26 + 17*x + 2*x^2;
P[1, x_] := 97 + 109*x + 29*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + x)*P[n - 1, x] + 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] = 26 + 17*x + 2*x^2; v[2] = 97 + 109*x + 29*x^2 + 2*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + x)*v[i - 1] + 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