Real-rooted atlas

A110441 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2628016158
34801200576642320
437161616013984358425637700
528768206384287232127488204801024671376
62227122559360539328036915201000960110592409612982520
7172416031190400959554569514854438748160709836857344016384270454912

All 15 rows

Technical details
Node id
n-3202fd904861a05e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-2745ba4cd7875635

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:         8,          4;
  2:        62,         80,          16;
  3:       480,       1200,         576,          64;
  4:      3716,      16160,       13984,        3584,         256;
  5:     28768,     206384,      287232,      127488,       20480,        1024;
  6:    222712,    2559360,     5393280,     3691520,     1000960,      110592,       4096;
  7:   1724160,   31190400,    95955456,    95148544,    38748160,     7098368,     573440,     16384;
  8:  13347856,  376062720,  1649262336,  2279409664,  1308548096,   353525760,   46735360,   2883584,    65536;
  9: 103334528, 4503763264, 27700537344, 51994945536, 40445296640, 15118716928, 2908487680, 290717696, 14155776, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (8+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) = 8 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 4*x;
P[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (8 + 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] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (8 + 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