Real-rooted atlas

A046716 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(18 + 96\,x + 128\,x^{2} - 9\,n - 48\,n x - 64\,n x^{2}\right)\,P_{n-2}(x) + \left(-2 - 8\,x + 3\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
24812
3197264155
411864810885122366
5102175201913618432409650205
612088115176413184667136425984327681666336
718372721851281038630424679936293847041418854426214481270487
83389242486606802907414409253416961654738944157705830462731059220971525129338050

All 15 rows

Technical details
Node id
n-268fc412659cc990 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-fc48a6a4812d9e4a

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:          4,           8;
   3:         19,          72,           64;
   4:        118,         648,         1088,           512;
   5:       1021,        7520,        19136,         18432,          4096;
   6:      12088,      115176,       413184,        667136,        425984,         32768;
   7:     183727,     2185128,     10386304,      24679936,      29384704,      14188544,         262144;
   8:    3389242,    48660680,    290741440,     925341696,    1654738944,    1577058304,      627310592,       2097152;
   9:   73156249,  1234033152,   8911969664,   35717423104,   85793583104,  123505999872,    98666283008,   33755758592,      16777216;
  10: 1804349548, 35015140296, 297031160832, 1438566079488, 4350542757888, 8412563079168, 10157104824320, 7000641503232, 2108896051200, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n-18+(48*n-96)*x+(64*n-128)*x^2)*P(n-2,x) + (3*n-2+(8*n-8)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(9*n - 18 + (48*n - 96)*x + (64*n - 128)*x^2)*P[n - 2, x] + (3*n - 2 + (8*n - 8)*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] = 4 + 8*x; for(n = 3, N, my(i = n); v[i] = -(9*n - 18 + (48*n - 96)*x + (64*n - 128)*x^2)*v[i - 2] + (3*n - 2 + (8*n - 8)*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