Real-rooted atlas

A091042 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 32\,x - 64\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 10 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2102434
376320320716
456833606272358413784
5424032256875529830436864259216
6316482942721047552175718413516803604484842784
7236224259584011435008255590403003187217039360340787290305216
8176320022356480117381120328048640520519680459669504201850880314572801683046784

All 15 rows

Technical details
Node id
n-38ea897b21ea55e6 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-68435dc840e803c3

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:       10,         24;
   3:       76,        320,         320;
   4:      568,       3360,        6272,        3584;
   5:     4240,      32256,       87552,       98304,        36864;
   6:    31648,     294272,     1047552,     1757184,      1351680,       360448;
   7:   236224,    2595840,    11435008,    25559040,     30031872,     17039360,      3407872;
   8:  1763200,   22356480,   117381120,   328048640,    520519680,    459669504,    201850880,    31457280;
   9: 13160704,  189120512,  1152827392,  3868196864,   7743078400,   9340715008,   6488588288,  2281701376,   285212672;
  10: 98232832, 1577687040, 10950770688, 42891608064, 103743750144, 159124553728, 152530059264, 86067118080, 24863834112, 2550136832;
%F Formula
The row polynomials satisfy P(n,x) = -(4+32*x+64*x^2)*P(n-2,x) + (8+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 10 + 24*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 10 + 24*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 32*x + 64*x^2)*P[n - 2, x] + (8 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 10 + 24*x; for(n = 3, N, my(i = n); v[i] = -(4 + 32*x + 64*x^2)*v[i - 2] + (8 + 16*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023